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

Abstract

We show that for a fixed $\mathrm{SL}_3(\mathbb Z)$ Hecke--Maass cusp form $F$, there are infinitely many primitive Dirichlet characters $\chi$ such that $L(1/2, F\times \chi)$ and $L(1/2, \chi)$ 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 $\{\chi\bmod q: q\in \mathcal Q\}$ where $\mathcal Q$ is a set of squarefree integers with a small divisor of suitable size.