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

Abstract Let 𝜋 and π ′ ^ be cuspidal automorphic representations of GL ⁢ (n) GL (n) and GL ⁢ (n ′) GL (n^) with unitary central characters. We establish a new zero-free region for all GL ⁢ (1) GL (1) -twists of the Rankin–Selberg 𝐿-function L ⁢ (s, π × π ′) L (s, ^), 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 ′ ⁢ (s, π × π ′) / L ⁢ (s, π × π ′) -L^ (s, ^) /L (s, ^). Also, for n ≤ 8 n 8, we extend the region of holomorphy and nonvanishing for the twisted symmetric power 𝐿-functions L ⁢ (s, π, Sym n ⊗ χ)