Value-distribution of cubic Hecke L-functions

Abstract

Let k=Q(−3), and let c∈Ok be a square free algebraic integer such that c≡1(mod〈9〉). Let ζk(c1/3)(s) be the Dedekind zeta function of the cubic field k(c1/3) and ζk(s) be the Dedekind zeta function of k. For fixed real σ>1/2, we obtain asymptotic distribution functions Fσ for the values of the logarithm and the logarithmic derivative of the Artin L-functionsLc(σ)=ζk(c1/3)(σ)ζk(σ), as c varies. Moreover, we express the characteristic function of Fσ explicitly as a product indexed by the prime ideals of Ok. As a corollary of our results, we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for the family of number fields {k(c1/3)}c. We also deduce a similar result for the Euler-Kronecker constants of this family.