Mixed moment and joint non-vanishing of $\mathrm{GL}(3)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)$ $L$-functions

Abstract

In this article, we study the mixed-moment problem for the product of central values of two $L$-functions twisted by a Dirichlet character of large conductor $q$. Precisely, for a fixed ${\mathrm{SL}}_3(\mathbb{Z})$ Hecke-Maass cusp form $\pi$, we asymptotically evaluate the first moment, with $\text{power-saving}$ error term for the product of central $L$-values $L(1/2,\pi \times \chi )\overline{L(1/2,\chi )}$ averaged over the $\text{full family}$ $\{\chi (\mathrm{mod}q):Q\le q\le 2Q\}$ of primitive Dirichlet characters modulo $q$. And as a consequence, we also established the simultaneous non-vanishing of the aforementioned central $L$-values.

arXiv commentSome errors in the earlier draft have been fixed. Comments are welcome!