Explicit lower bounds on $|L(1, χ)|$

Abstract

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q, and let L(s, χ) denote its associated Dirichlet L-function. We show that |L(1, χ)| ≥ 1/(9.12255 log(q/π)) for sufficiently large q, and that |L(1, χ)| ≥ 1/(9.69030 log(q/π)) for all q ≥ 2, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.