Simultaneous non-vanishing of central values of $\mathrm{GL}(2)\times \mathrm{GL}(3)$ and $\mathrm{GL}(3)\times \mathrm{GL}(3)$ $L$-functions

Abstract

Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k\equiv 0(\mathrm{mod}4)$ on ${\mathrm{SL}}_2(\mathbb{Z})$, and let $\pi$ be a fixed cuspidal automorphic representation of ${\mathrm{GL}}_3$. In this paper, we establish an asymptotic formula for the first moment of the product \[L(1/2,g\times F)L(1/2,π\times F),\] where $F$ runs over an orthonormal basis of Hecke-Maaß cusp forms of level $q$ on ${\mathrm{GL}}_3$. As an application, we deduce that $L(1/2,g\times F)L(1/2,\pi \times F)\ne 0$ for infinitely many such forms $F$.

14 pages