Explicit bounds for L-functions on the edge of the critical strip

Abstract

Assuming GRH and the Ramanujan–Petersson conjecture we prove explicit bounds for L(1,f) for a large class of L-functions L(s,f), which includes L-functions attached to automorphic cuspidal forms on GL(n). The proof generalizes work of Lamzouri, Li and Soundararajan. Furthermore, the main results improve the classical bounds of Littlewood(1+o(1))(12eγπ2log⁡log⁡C(f))−d≤|L(1,f)|≤(1+o(1))(2eγlog⁡log⁡C(f))d, where C(f) is the analytic conductor of L(s,f).