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

Abstract

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 $\frac{1}{2}<\sigma<1$, define $m(\sigma)$ as the supremum of all numbers $m$ such that \[ \int_{1}^T|L(\sigma+it, \text{sym}^2 f)|^m \text{d}t\ll_f T^{1+\varepsilon}, \] where $\epsilon>0$ is an arbitrarily small number. In this paper, we established the bound \begin{align*} m(\sigma)\geq \frac{17}{26-28\sigma}, \text{ for }\frac{5}{8}\leq\sigma\leq\frac{52}{73}, \end{align*} which improved our previous result.