Open Access
2016 Lower bound for the higher moment of symmetric square $L$-functions
Guanghua Ji
Rocky Mountain J. Math. 46(3): 915-923 (2016). DOI: 10.1216/RMJ-2016-46-3-915


Let $\mathcal {S}_k(N)$ be the space of holomorphic cusp forms of weight $k$, level $N$ and let $\mathcal {B}_k(N)$ be an orthogonal basis of $\mathcal {S}_k(N)$ consisting of newforms. Let $L(s, \textup {sym}^2 f)$ be the symmetric square $L$-function of $f\in \mathcal {B}_k(N)$. In this paper, the lower bound of the higher moment of $L(1/2,\textup {sym}^2 f)$ is established, i.e., for any even positive number $r$, \[ \sum _{f\in \mathcal {B}_k(N)}\omega _f^{-1}L\bigg (\frac 12, \textup {sym}^2 f\bigg )^r \gg (\log N)^{{r(r+1)}/{2}} \] holds for $N\rightarrow \infty $.


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Guanghua Ji. "Lower bound for the higher moment of symmetric square $L$-functions." Rocky Mountain J. Math. 46 (3) 915 - 923, 2016.


Published: 2016
First available in Project Euclid: 7 September 2016

zbMATH: 06628759
MathSciNet: MR3544839
Digital Object Identifier: 10.1216/RMJ-2016-46-3-915

Primary: 11F11 , 11F66

Keywords: Holomorphic forms , lower bound , symmetric square $L$-functions

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 3 • 2016
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