Lower Bounds for Rankin-Selberg $L$-functions on the Edge of the Critical Strip

Abstract

Let $F$ be a number field, and let $\pi_1$ and $\pi_2$ be distinct unitary cuspidal automorphic representations of $\operatorname{GL}_{n_1}(\mathbb{A}_F)$ and $\operatorname{GL}_{n_2}(\mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, \pi_1 \times \widetilde{\pi}_2)$ along the edge $\Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, \pi_1 \times \widetilde{\pi}_2)$ is also determined.