Explicit Bounds for $L$-Functions on the Edge of the Critical Strip

TL;DR

This paper establishes explicit bounds for the values of certain $L$-functions at 1, assuming GRH and Ramanujan-Petersson, extending previous work and improving classical bounds related to the analytic conductor.

Contribution

It generalizes existing bounds for $L$-functions at 1 to a broad class including automorphic forms on $GL(n)$ under GRH and Ramanujan-Petersson assumptions.

Findings

Derived explicit bounds for $L(1,f)$ under GRH and Ramanujan-Petersson.

Extended bounds to automorphic $L$-functions on $GL(n)$.

Improved classical bounds involving the analytic conductor.

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 \[(1+o(1))\left(\frac{12e^{\gamma}}{\pi^2}\log\log C(f)\right)^{-d} \leq |L(1,f)|\leq (1+o(1))\Big(2e^{\gamma}\log\log C(f)\Big)^d,\] where $C(f)$ is the analytic conductor of $L(s,f)$.