Simultaneous non-vanishing of twists of ${\mathrm{G}\mathrm{L}}_3$ and ${\mathrm{G}\mathrm{L}}_1$ $L$-functions

We show that for a fixed ${\mathrm{S}\mathrm{L}}_3(\mathbb{Z})$ Hecke–Maass cusp form $F$, there are infinitely many primitive Dirichlet characters $\chi$ such that $L(1\mathrm{/}2,F\times \chi )$ and $L(1\mathrm{/}2,\chi )$ do not vanish \emph{simultaneously}. More precisely, we obtain an asympotic formula with a {\em power-saving} error term for the product of these central values averaged over $\{\chi q:q\in \mathcal{Q}\}$ where $\mathcal{Q}$ is a set of squarefree integers with a small divisor of suitable size.