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

Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k \equiv 0 \pmod{4}$ 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 \[L(1/2,g\times F)L(1/2,\pi\times F),\] where $F$ runs over an orthonormal basis of Hecke-Maa{\ss} cusp forms of level $q$ on $\mathrm{GL}_3$. As an application, we deduce that $L(1/2, g\times F)L(1/2,\pi\times F)\neq 0$ for infinitely many such forms $F$.