Numerical estimates on the Landau-Siegel zero and other related quantities

Abstract

A. Let 𝑞 be an odd prime, 𝜒 be a non-principal Dirichlet character mod 𝑞 and 𝐿(𝑠, 𝜒) be the associated Dirichlet 𝐿-function. Exploiting a fast algorithm to compute the values of |𝐿 (1, 𝜒)| for every odd prime 𝑞 ≤ 10 7 , we show that 𝐿(1, 𝜒 ) > 𝑐 1 log 𝑞 and 𝛽 < 1 − 𝑐 2 log 𝑞 , where 𝑐 1 = 0.0124862668 . . . , 𝑐 2 = 0.0091904477 . . . , 𝜒 is the quadratic Dirichlet character mod 𝑞 and 𝛽 ∈ (0, 1), if exists, is the Landau-Siegel zero of the set of such Dirichlet 𝐿-functions. As a by-product of the computations here performed, we also obtained some information about Littlewood's and Joshi's bounds on 𝐿 (1, 𝜒 ) and on the class number of the imaginary quadratic field ℚ( √ −𝑞).