Higher moments of the symmetric square $L$-function off the critical line
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917432923258880 |
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| author | Wang, You Jun |
| author_facet | Wang, You Jun |
| contents | Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$
be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<σ<1$, define
$m(σ)$ as the supremum of all numbers $m$ such that
\[
\int_{1}^T|L(σ+it, \text{sym}^2 f)|^m \text{d}t\ll_f T^{1+\varepsilon},
\]
where $ε>0$ is an arbitrarily small number. In this paper, we established the bound
\begin{align*}
m(σ)\geq \frac{17}{26-28σ}, \text{ for }\frac{5}{8}\leqσ\leq\frac{52}{73},
\end{align*}
which improved our previous result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22272 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher moments of the symmetric square $L$-function off the critical line Wang, You Jun Number Theory 11F11 11F66 Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<σ<1$, define $m(σ)$ as the supremum of all numbers $m$ such that \[ \int_{1}^T|L(σ+it, \text{sym}^2 f)|^m \text{d}t\ll_f T^{1+\varepsilon}, \] where $ε>0$ is an arbitrarily small number. In this paper, we established the bound \begin{align*} m(σ)\geq \frac{17}{26-28σ}, \text{ for }\frac{5}{8}\leqσ\leq\frac{52}{73}, \end{align*} which improved our previous result. |
| title | Higher moments of the symmetric square $L$-function off the critical line |
| topic | Number Theory 11F11 11F66 |
| url | https://arxiv.org/abs/2604.22272 |