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Moments of L-functions attached to the twist of modular form by Dirichlet characters

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Abstract

Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL 2(ℤ). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for \(k = \tfrac{1} {n}\) with n ∈ ℕ,

$$M_k \left( {q,f} \right) = \sum\limits_{\begin{array}{*{20}c} {\chi (\bmod q)} \\ {\chi \ne \chi _0 } \\ \end{array} } {\left| {L\left( {\frac{1} {2},f \otimes \chi } \right)} \right|^{2k} < < _k \varphi \left( q \right)(\log q)^{k^2 } ,}$$

and the result also holds for any real number 0 < k < 1 under the GRH for L(s, f ⊗ χ). The authors also prove that under the GRH for L(s, f ⊗ χ),

$$M_k \left( {q,f} \right) > > _k \varphi (q)(log q)^{k^2 }$$

for any real number k > 0 and any large prime q.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Guanghua Ji

  2. School of Mathematics and Statistics, Shandong University, Weihai, 264209, Shandong, China

    Haiwei Sun

Authors
  1. Guanghua Ji
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  2. Haiwei Sun
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Correspondence to Haiwei Sun.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11301299), the Natural Science Foundation of Shandong Province (No. ZR2012AQ001) and the Specialized Research Fund for the Doctoral Program of Higher Education (New Teachers) (Nos. 20110131120001, 20120131120075).

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Ji, G., Sun, H. Moments of L-functions attached to the twist of modular form by Dirichlet characters. Chin. Ann. Math. Ser. B 36, 237–252 (2015). https://doi.org/10.1007/s11401-015-0886-8

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  • DOI: https://doi.org/10.1007/s11401-015-0886-8

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