Two standard hypothesis. We introduce two standard hypothe- Dirichlet coefficients of the symmetric power L-functions. In

Special values of symmetric power L-functions and Hecke eigenvalues 719 n. Moreover, if N is another integer, then we decompose n as n = n N n N with p | n N ⇒ p | N and n N , N = 1. The functions ✶ N and ✶ N are defined by ✶ N n := 1 if the prime divisors of n divide N 0 otherwise and ✶ N n := 1 if n, N = 1 0 otherwise. The letters s and ρ are devoted to complex numbers and we set ℜe s = σ and ℜe ρ = r.

2. Modular tools

In this section, we establish some results needed for the forthcoming proofs of our results.

2.1. Two standard hypothesis. We introduce two standard hypothe-

sis that shall allow us to prove our results for each symmetric power L– function. If f ∈ H ∗ 2 N , we have defined Ls, Sym m f in 6 as being an Euler product of degree m + 1. These representations allow to express the multiplicativity relation of n 7→ λ f n: this function is multiplicative and, if p ∤ N and ν ≥ 0, we have 30 λ f p ν = χ Sym ν [gθ f,p ]. Recall also that n 7→ λ f n is strongly multiplicative on integers having their prime factors in the support of N and that if n | N, then 31 |λ f n | = 1 √ n . The first hypothesis on the automorphy of Ls, Sym m f for all f ∈ H ∗ 2 N is denoted by Sym m N . It is has been proved in the cases m ∈ {1, 2, 3, 4} see [GJ78], [KS02b], [KS02a] and [Kim03]. The second hypothesis is con- cerned with the eventual Landau-Siegel zero of the mth symmetric power L- functions, it is denoted by LSZ m N and has been proved for m ∈ {1, 2, 4} see [HL94], [GHL94], [HR95] and [RW03]. Fix m ≥ 1 and N a squarefree positive integer. Hypothesis Sym m N . For every f ∈ H ∗ 2 N , there exists an automorphic cuspidal selfdual representation of GL m+1 A Q whose local L factors agree 720 Emmanuel Royer, Jie Wu with the ones of the function Ls, Sym m f . Define L ∞ s, Sym m f :=                      π −s2 Γ s 2 2 u u Y j=1 2π −s−j Γ s + j if m = 2u with u even π −s+12 Γ s + 1 2 2 u u Y j=1 2π −s−j Γ s + j if m = 2u with u odd 2 u+1 u Y j=0 2π −s−j−12 Γ s + j + 1 2 if m = 2u + 1. Then there exists εSym m f ∈ {−1, 1} such that N ms2 L ∞ s, Sym m f Ls, Sym m f = εSym m f N m1 −s2 L ∞ 1 − s, Sym m f L1 − s, Sym m f . We refer to [CM04] for a discussion on the analytic implications of this conjecture. The second hypothesis we use is the non existence of Landau- Siegel zero. Let N squarefree such that hypothesis Sym m N holds. Hypothesis LSZ m N . There exists a constant A m 0 depending only on m such that for every f ∈ H ∗ 2 N , Ls, Sym m f has no zero on the real interval [1 − A m log2N , 1].

2.2. Dirichlet coefficients of the symmetric power L-functions. In

this section, we study the Dirichlet coefficients of Ls, Sym m f z . We derive our study from the one of Cogdell Michel but try to be more explicit in our specific case. We begin with the polynomial D introduced in 4. Since Sym m is selfdual, we have, DX, Sym m , g ∈ R[X] and for x ∈ [0, 1[, 32 1 + x −m−1 ≤ Dx, Sym m , g ≤ 1 − x −m−1 . Remark 1. Note that the upper bound is optimal since the equation Sym m g = I admits always I as a solution whereas the lower bound is optimal only for odd m since Sym m g = −I has a solution only for odd m. Evaluating 32 at g = gπ, we find min g ∈SU2 D X, Sym 2m+1 , g = 1 + X −2m−2 . Special values of symmetric power L-functions and Hecke eigenvalues 721 Next, D h X, Sym 2m , g π 2m i = 1 − X −1 m Y j=1 1 − Xe 2j πi 2m −1 1 − Xe −2j πi 2m −1 = 1 + X −1 1 − X 2m −1 so that min g ∈SU2 D X, Sym 2m , g ≤ 1 + X −1 1 − X 2m −1 . For every g ∈ SU2, define λ z,ν Sym m g by the expansion 33 DX, Sym m , g z =: + ∞ X ν=0 λ z,ν Sym m gX ν . The function g 7→ λ z,ν Sym m g is central so that it may be expressed as a linear combination of the characters of irreducible representations of SU2. These characters are defined on the conjugacy classes of SU2 by 34 χ Sym m [gθ] = tr Sym m [gθ] = sin[m + 1θ] sin θ = X m 2 cos θ where X m is the mth Chebyshev polynomial of second kind on [ −2, 2]. We then have 35 λ z,ν Sym m g = X m ′ ≥0 µ z,ν Sym m ,Sym m′ χ Sym m′ g with µ z,ν Sym m ,Sym m′ = Z SU2 λ z,ν Sym m gχ Sym m′ g dg 36 = 2 π Z π λ z,ν Sym m [gθ] sin[m ′ + 1θ] sin θ dθ. 37 We call µ z,ν Sym m ,Sym m′ the harmonic of λ z,ν Sym m of order m ′ . In particular, 38 µ z,0 Sym m ,Sym m′ = δm ′ , 0 and, since λ z,1 Sym m g = zχ Sym m g, we have 39 µ z,1 Sym m ,Sym m′ = zδm, m ′ . From the expansion 40 1 − x −z = + ∞ X ν=0 z + ν − 1 ν x ν 722 Emmanuel Royer, Jie Wu we deduce D[x, Sym m , gθ] z = + ∞ X ν=0          X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 z + ν j+1 − 1 ν j+1   e iℓm,νθ          x ν with 41 ℓm, ν := mν − 2 m X k=1 kν k+1 and gets 42 λ z,ν Sym m [gθ] = X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 z + ν j+1 − 1 ν j+1   e iℓm,νθ . This function is entire in z, then assuming that z in real, using that the left hand side is real in that case, taking the real part in the right hand side and using analytic continuation we have for all z complex 43 λ z,ν Sym m [gθ] = X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 z + ν j+1 − 1 ν j+1   cos [ℓm, νθ] . It follows that 37 may be rewritten as µ z,ν Sym m ,Sym m′ = 2 π X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 z + ν j+1 − 1 ν j+1   × Z π cos [ℓm, νθ] sin[m ′ + 1θ] sin θ dθ that is 44 µ z,ν Sym m ,Sym m′ = 1 2 X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 z + ν j+1 − 1 ν j+1   ∆m, m ′ , ν with 45 ∆m, m ′ , ν =          2 if ℓm, ν = 0 and m ′ = 0 1 if ℓm, ν ± m ′ = 0 and m ′ 6= 0 −1 if ℓm, ν ± m ′ = ∓2 otherwise. Special values of symmetric power L-functions and Hecke eigenvalues 723 In particular, µ z,ν Sym m ,Sym m′ = 0 if m ′ mν thus 46 λ z,ν Sym m g = mν X m ′ =0 µ z,ν Sym m ,Sym m′ χ Sym m′ g. Equation 45 also immediately gives 47 µ z,ν Sym 2m ,Sym 2m′+1 = 0 and µ z,ν Sym 2m+1 ,Sym m′ = 0 if m ′ and ν have different parity for all m and m ′ . For m = 1, we have 48 D[X, St, gθ] = 1 1 − 2 cosθX + X 2 = + ∞ X ν=0 X ν 2 cos θX ν hence λ 1,ν St g = χ Sym ν g for all g ∈ SU2. It follows that 49 µ 1,ν St,Sym ν′ = δν, ν ′ . Now, equation 43 implies λ z,ν Sym m [gθ] ≤ X ν ∈Z m+1 ≥0 tr ν=ν   m Y j=0 |z| + ν j+1 − 1 ν j+1   = λ |z|,ν Sym m [g0] and + ∞ X ν=0 λ |z|,ν Sym m [g0]X ν = det[I − X Sym m g0] −|z| = 1 − X −m+1|z| so that 50 |λ z,ν Sym m [gθ] | ≤ m + 1 |z| + ν − 1 ν . From 45, remarking that the first case is incompatible with the second and third ones, that the two cases in the second case are incompatible and that the two cases of the third case are incompatible, we deduce that mν X m ′ =0 ∆m, m ′ , ν ≤ 2 and 44 gives 51 mν X m ′ =0 µ z,ν Sym m ,Sym m′ ≤ m + 1 |z| + ν − 1 ν . 724 Emmanuel Royer, Jie Wu This is a slight amelioration of Proposition 2.1 of [CM04] in the case of SU2. It immediately gives µ z,ν Sym m ,Sym m′ ≤ m + 1 |z| + ν − 1 ν . 52 To conclude this study, define the multiplicative function n 7→ λ z Sym m f n by the expansion Ls, Sym m f z =: + ∞ X n=1 λ z Sym m f nn −s . 53 For easy reference, we collect the results of the previous lines in the Proposition 2. Let N be a squarefree integer, f ∈ H ∗ 2 N ; let ν ≥ 0 and m 0 be integers and z be a complex number. Then λ z Sym m f p ν =              τ z p ν λ f p mν if p | N mν X m ′ =0 µ z,ν Sym m ,Sym m′ λ f p m ′ if p ∤ N . Moreover, λ z Sym m f p ν ≤ τ m+1 |z| p ν µ 1,ν St,Sym ν′ = δν, ν ′ µ z,0 Sym m ,Sym m′ = δm ′ , 0 µ z,1 Sym m ,Sym m′ = zδm, m ′ µ z,ν Sym 2m ,Sym 2m′+1 = 0 µ z,ν Sym 2m+1 ,Sym m′ = 0 if m ′ and ν have different parity, and mν X m ′ =0 µ z,ν Sym m ,Sym m′ ≤ m + 1 |z| + ν − 1 ν . Proof. We just need to prove the first equation. Assume that p | N, then ∞ X ν=0 λ z Sym m f p ν p −νs = [1 − λ f p ν p −s ] −z and the result follows from 40 since n 7→ λ f n is strongly multiplicative on integers having their prime factors in the support of N . In the case Special values of symmetric power L-functions and Hecke eigenvalues 725 where p ∤ N , we have ∞ X ν=0 λ z Sym m f p ν p −νs = D[p −s , Sym m , gθ f,p ] −z so that the results are consequences of λ z Sym m f p ν = λ z,ν Sym m [gθ f,p ] and especially of 46 and 30. We shall need the Dirichlet series 54 W z,ρ m,N s = + ∞ X n=1 ̟ z,ρ m,N n n s where ̟ z,ρ m,N is the multiplicative function defined by 55 ̟ z,ρ m,N p ν =      if p | N mν X m ′ =0 µ z,ν Sym m ,Sym m′ p ρm ′ otherwise for all prime number p and ν ≥ 1. Similarly, define a multiplicative function e w z,ρ m,N by 56 e w z,ρ m,N p ν =      if p | N mν X m ′ =0 |µ z,ν Sym m ,Sym m′ | p ρm ′ otherwise. Using equations 39 and 52, we have 57 + ∞ X ν=0 e w z,ρ m,N p ν p σν ≤ 1 − 1 p σ −m+1|z| − m + 1 |z| p σ + m + 1 |z| p σ+r 1 − 1 p σ −m+1|z|−1 so that the series converges for ℜe s 12 and ℜe s+ℜe ρ 1. We actually have an integral representation. Lemma 3. Let s and ρ in C such that ℜe s 12 and ℜe s + ℜe ρ 1. Let N be squarefree, then W z,ρ m,N s = Y p∤N Z SU2 Dp −s , Sym m , g z Dp −ρ , St, g dg. 726 Emmanuel Royer, Jie Wu Moreover, W z,ρ 2m,N s = 1 ζ N 4ρ Y p∤N Z SU2 Dp −s , Sym 2m , g z Dp −2ρ , Sym 2 , g dg. Remark 4. The key point of Corollary E is the fact that the coefficients ap- pearing in the series expansion of DX, Sym 2m , g have only even harmonics – see equations 46 and 47. This allows to get the second equation in Lemma 3. It does not seem to have an equivalent for DX, Sym 2m+1 , g. Actually, we have W z,ρ 2m+1,N s = Y p∤N Z SU2 [1 − p −4ρ + p −ρ 1 − p −2ρ χ St g] × Dp −s , Sym 2m+1 , g z Dp −2ρ , Sym 2 , g dg and the extra term p −ρ 1 − p −2ρ χ St g is the origin of the fail in obtaining Corollary E for odd powers. Before proving Lemma 3, we prove the following one Lemma 5. Let g ∈ SU2, ℓ ≥ 2 an integer and |X| 1. Then + ∞ X k=0 χ Sym k gX k = DX, St, g and + ∞ X k=0 χ Sym kℓ gX k = [1 + χ Sym ℓ−2 gX]DX, St, g ℓ . In addition, + ∞ X k=0 χ Sym 2k gX k = 1 − X 2 DX, Sym 2 , g. Proof. Let g ∈ SU2. Denote by e iθ and e −iθ its eigenvalues. The first point is equation 48. If ℓ ≥ 2, with ξ = exp2πiℓ, λ = e iθ and x = 2 cos θ we have + ∞ X ν=0 X ℓν xt ℓν = 1 ℓ ℓ −1 X j=0 1 1 − λξ j t1 − λξ j t . On the other hand, ℓ −1 X j=0 1 1 − λξ j t = ℓ −1 X j=0 + ∞ X n=0 λ n ξ jn t n = ℓ 1 − λ ℓ t ℓ Special values of symmetric power L-functions and Hecke eigenvalues 727 so that + ∞ X ν=0 X ℓν xt ν = 1 + λ ℓ −1 − λ ℓ −1 λ − λ t 1 − λ ℓ + λ ℓ t + t 2 . Since λ ℓ −1 − λ ℓ −1 λ − λ = X ℓ −2 x we obtain the announced result. In the case ℓ = 2, it leads to + ∞ X k=0 χ Sym 2k gt k = 1 + t 1 − λ 2 t1 − λ 2 t = 1 − t 2 Dt, Sym 2 , g. Proof of Lemma 3. It follows from mν X m ′ =0 µ z,ν Sym m ,Sym m′ p ρm ′ = + ∞ X m ′ =0 µ z,ν Sym m ,Sym m′ p ρm ′ and the expression 36 that W z,ρ m,N s = Y p∤N Z SU2 + ∞ X ν=0 λ z,ν Sym m g p νs + ∞ X m ′ =0 χ Sym m′ g p m ′ ρ dg. The first result is then a consequence of Lemma 5. Next, we deduce from 47 that W z,ρ 2m,N s = Y p∤N + ∞ X ν=0 1 p νs + ∞ X m ′ =0 µ z,ν Sym 2m ,Sym 2m′ p 2ρm ′ and the second result is again a consequence of Lemma 5. We also prove the Lemma 6. Let m ≥ 1. There exists c 0 such that, for all N squarefree, z ∈ C, σ ∈ ]12, 1] and r ∈ [12, 1] we have X n ≥1 e w z,ρ m,N n n s ≤ exp cz m + 3 log 2 z m + 3 + z m + 3 1 −σσ − 1 1 − σ logz m + 3 where 58 z m := m + 1 min{n ∈ Z ≥0 : n ≥ |z|}. 728 Emmanuel Royer, Jie Wu Proof. Equation 57 gives Y p σ ≤z m +3 X ν ≥0 1 p νσ X ≤ν ′ ≤mν µ z,ν Sym m ,Sym ν′ p rν ′ ≤ Y p σ ≤z m +3 1 − 1 p σ −z m −1 1 + z m p σ+12 . Using X p ≤y 1 p σ ≤ log 2 y + y 1 −σ − 1 1 − σ log y valid uniformely for 12 ≤ σ ≤ 1 and y ≥ e 2 see [TW03, Lemme 3.2] we obtain Y p σ ≤z m +3 + ∞ X ν=0 e w z,r m,N p ν p νσ ≤ exp cz m + 3 log 2 z m + 3 + z m + 3 1 −σσ − 1 1 − σ logz m + 3 . For p σ z m + 3, again by 57, we have X ν ≥0 1 p νσ X ≤ν ′ ≤mν µ z,ν Sym m ,Sym ν′ p rν ′ ≤ 1 + cz m + 3 2 p 2σ + cz m + 3 p σ+12 , so that Y p σ z m +3 + ∞ X ν=0 e w z,r m,N p ν p νσ ≤ e cz m +3 1σ logz m +3 ≤ exp cz m + 3 z m + 3 1 −σσ − 1 1 − σ logz m + 3 . For the primes dividing the level, we have the Lemma 7. Let ℓ, m ≥ 1. For σ ∈ ]12, 1] and r ∈ [12, 1] we have Y p |N Z SU2 Dp −s , Sym m , g z Dp −ρ , Sym ℓ , g dg = 1 + O m,ℓ Err with Err := ωN P − N 2r + |z|ωN P − N r+σ + |z| 2 ωN P − N 2σ Special values of symmetric power L-functions and Hecke eigenvalues 729 uniformely for N ∈ N max ω · 12r , [ |z|ω·] 1r+σ , [ |z| 2 ω ·] 12σ , z ∈ C. Proof. Write Ψ z m,ℓ p := Z SU2 Dp −s , Sym m , g z Dp −ρ , Sym ℓ , g dg. Using 35 and the orthogonality of characters, we have Ψ z m,ℓ p = + ∞ X ν 1 =0 + ∞ X ν 2 =0 p −ν 1 s −ν 2 ρ minmν 1 ,ℓν 2 X ν=0 µ z,ν 1 Sym m ,Sym ν µ 1,ν 2 Sym ℓ ,Sym ν . Proposition 2 gives Ψ z m,ℓ p − 1 ≤ + ∞ X ν 2 =2 ν 2 + ℓ ν 2 1 p rν 2 + |z| p σ + ∞ X ν 2 =1 ν 2 + ℓ ν 2 1 p rν 2 + + ∞ X ν 1 =2 m + 1 |z| + ν 1 − 1 ν 1 1 p σν 1 + ∞ X ν 2 =0 ν 2 + ℓ ν 2 1 p rν 2 ≪ m,ℓ 1 p 2r + |z| p r+σ + |z| 2 p 2σ which leads to the result. Using 49 we similarly can prove the Lemma 8. Let m ≥ 1. Then we have Z SU2 Dp −1 , Sym m , g z Dp −12 , St, g dg = 1 + O m |z| p 1+m2 uniformely for z ∈ C and p ≥ m + 1|z| + 3.

2.3. Dirichlet coefficients of a product of L-functions. The aim of