Abstract
Let S k (N, χ) be the space of cusp forms of weight k, level N and character χ. For \(f \in S_k(N, \chi)\) let L(s, sym2 f) be the symmetric square L-function and \(L(s, f \otimes f)\) be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym2 f) and \(L(s, f \otimes f)\) over a basis of S k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym2 f) does not vanish.
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References
Blomer, V.: Nonvanishing of class group L-functions at the central point. Ann. Inst. Fourier 54, 831–847 (2004)
Conrey, B., Iwaniec, H.: The cubic moment of central values of automorphic L-functions. Ann. Math. 151, 1175–1216 (2002)
Delange, H.: Généralisation du théorème de Ikehara. Ann. Sci. Ecole Norm. Sup. 71, 213–242 (1954)
Duke, W.: The critical order of vanishing of automorphic L-functions with large level. Invent. Math. 119, 165–174 (1995)
Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero. Ann. Math. 140, 161–181 (1994)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society, Providence (2004)
Iwaniec, H., Michel, P.: The second moment of the symmetric square L-function. Ann. Acad. Sci. Fenn. Math. 26, 465–482 (2001)
Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of L-functions. Geom. Funct. Anal., Special Volume, pp. 705-741 (2000)
Iwaniec, H., Sarnak, P.: The non-vanishing of central values of automorphic L-functions and the Landau-Siegel zero. Isr. J. Math. 120, 155–177 (2000)
Kohnen, W., Sengupta, J.: On the average of central values of symmeric square L-functions in weight aspect. Nagoya Math. J. 167, 95–100 (2002)
Kowalski, E., Michel, P.: Zeros of families of automorphic L-functions close to 1. Pac. J. Math. 207, 411–431 (2002)
Kowalski, E., Michel, P., VanderKam, J.: Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142, 95–151 (2000)
Lau, Y.-K.: An estimate for symmetric square L-functions on weight aspect. Arch. Math. 81, 169–174 (2003)
Li, W.: L-series of Rankin type and their functional equation. Math. Ann. 244, 135–166 (1979)
Murty, M.R., Murty, V.K.: Non-vanishing of L-functions and applications, Progress in Mathematics, vol. 157. Birkhäuser, Basel (1997)
Oberhettinger, F.: Tables of Mellin Transforms. Springer, Heidelberg (1974)
Rankin, R.A.: Contributions the theory of Ramanujan’s function τ(n) and similar arithmetical functions II. Proc. Camb. Philol. Soc. 35, 357–372 (1939)
Royer, E.: Statistique de la variable aléatoire L(Sym2f, 1). Math. Ann. 321, 667–687 (2001)
Royer, E., Wu, J.: Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique. Rev. Iberoam. 21, 263–312 (2005)
Sankaranarayanan, A.: Fundamental properties of symmetric square L-functions I. Ill. J. Math. 46, 23–43 (2002)
Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. 31, 79–98 (1975)
Soundararajan, K.: Non-vanishing of quadratic Dirichlet L-functions at \(s = \frac{1}{2}\). Ann. Math. 152(2), 447–488 (2000)
Zagier, D.: Zetafunktionen und quadratische Zahlkörper. Springer, Heidelberg (1981)
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The author was supported by NSERC grant 311664-05.
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Blomer, V. On the central value of symmetric square L-functions. Math. Z. 260, 755–777 (2008). https://doi.org/10.1007/s00209-008-0299-4
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DOI: https://doi.org/10.1007/s00209-008-0299-4