Non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$

Shingo Sugiyama, Masao Tsuzuki

Published 2018-05-01Version 1

Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$, $f$ and $\bar f$, where $f$ runs over the normalized Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $l \ge 4$. As an application, we prove an infinitude of pairs $(\phi, F)$ of $\phi$ as above and a cohomological cusp form $F$ on ${\rm SL}_3(\mathbb{Z})$ such that $L(1/2, \phi \times F)= 0$.