OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re

The theorem states that, for the field K = ℚ(ζ₃) with ring of integers O (ω being the cyclotomic generator of O), any Hecke-type character χ, and any complex number s with Re(s) > 7/8, the L-function L(χ,s) is nonzero, except possibly at the pole case where χ.residue is the trivial multiplicative character 1 and s = 1. Here a character χ consists of a nonzero ideal modulus 𝔪 of O, a multiplicative character of (O/𝔪)ˣ valued in ℂ (called residue) that is trivial on the images of all units of O, and a positive integer period N with N in 𝔪. Its coefficients are indexed by pairs (x,y) in {0,…,N−1}², with value the residue character at the class of x + yω modulo 𝔪. L(χ,s) is defined as (1/6) · π^s · Γ(s)⁻¹ times the completed lattice function Λ(s) built from these coefficients through a weak functional-equation pair. The hypothesis excludes exactly the case where the residue character equals 1 and s = 1. The stated result is admitted in the source, not proved there.