Abstract
Let {ϕ j (z) : j ≥ 1} be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue 1/4 + t 2 j . For each ϕ j (z), we have the automorphic L-function L(s, sym2 ϕ j ), which is called the symmetric square L-function associated to ϕ j . In this paper, we consider the average estimate of L(1/2, sym2 ϕ j ) and prove that, for sufficiently large T, the estimate \( {\displaystyle {\sum}_{T<{t}_j\le T+M}{\left|L\left(1/2,{\mathrm{sym}}^2{\phi}_j\right)\right|}^2\ll {T}^{1+\varepsilon }M} \) holds for T 1/3 + ε ≤ M ≤ T 1 − ε.
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(Hengcai Tang) The author is supported by the project of the National Natural Science Foundation of China (11301142) and the key project of colleges and universities of Henan Province (15A110014).
(Zhao Xu) The author is supported by the project of the National Natural Science Foundation of China (11501327) and the Fundamental Research Funds of Shandong University (2014GN027).
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Tang, H., Xu, Z. Central value of the symmetric square L-functions related to Hecke–Maass forms. Lith Math J 56, 251–267 (2016). https://doi.org/10.1007/s10986-016-9317-0
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DOI: https://doi.org/10.1007/s10986-016-9317-0