A new zero-free region for Rankin-Selberg $L$-functions

Abstract

Let $\pi$ and ${\pi }^{\prime }$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n^{\prime })$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\prime })$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L^{\prime }(s,\pi \times {\pi }^{\prime })/L(s,\pi \times {\pi }^{\prime })$. Also, for $n\le 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,\pi ,{\mathrm{Sym}}^n\otimes \chi )$ of any cuspidal automorphic representation of $\mathrm{GL}(2)$.

21 pages, LaTeX2e; v2: updated Section 3 and the Remark below Theorem 1.1; v3: updated Section 1, added Theorem 2.2 and Section 8, added the Remark below the proof of Proposition 4.4; v4: updated the end of Section 1, added Section 4 (part of which was formerly Section 5.2); v5: updated Section 1; v6: updated Section 5

References in corpus (5)

• Uniform approximate functional equation for principal L-functions
• Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve Theory
• Weak subconvexity without a Ramanujan hypothesis
• Towards a ${\mathrm{GL}}_n$ variant of the Hoheisel phenomenon
• Zeros of Rankin-Selberg $L$-functions in families

Cited by in corpus (1)

• Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions