Higher moments of the symmetric square $L$-function off the critical line

Abstract

Let $f$ be the Hecke eigenform for the modular group $S{L}_2(\mathbb{Z})$, and $L(s,{\text{sym}}^{2}f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma <1$, define $m(\sigma )$ 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 $\epsilon >0$ 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.

8 pages