A note on standard zero-free regions for Rankin–Selberg $L$-functions
Volume 196 / 2020
Acta Arithmetica 196 (2020), 423-431
MSC: Primary 11M26; Secondary 11F66.
DOI: 10.4064/aa190825-1-2
Published online: 31 August 2020
Abstract
Let $ \pi $ be a unitary cuspidal automorphic representation of $\mathrm{GL}_{n}(\mathbb{A}_{F}) $ and $ \tilde {\pi } $ its contragredient. In this short note, we prove a standard zero-free region in the $ t $-aspect for the Rankin–Selberg $ L $-function $ L(s,\pi \times \tilde {\pi }) $, assuming that the fourth powers of the Fourier coefficients $ \lambda _\pi (\mathfrak {p}) $ of $ \pi $ at primes $ \mathfrak {p} $ are bounded on average. For $ n=3 $, we prove the desired fourth moment bound under the assumption of the existence of Langlands $ \operatorname{Sym} ^2 $ lift from $ \mathrm {GL}_3 $ to $ \mathrm {GL}_6 $.
