Value-distribution of cubic Hecke $L$-functions

Abstract

Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in {\mathfrak{O}}_k$ be a square free algebraic integer such that $c\equiv 1(\mathrm{mod}⟨9⟩)$. Let ${\zeta }_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$ and ${\zeta }_k(s)$ be the Dedekind zeta function of $k$. For fixed real $\sigma >1/2$, we obtain asymptotic distribution functions $F_{\sigma }$ for the values of the logarithm and the logarithmic derivative of the Artin $L$-functions \begin{equation*} L_c(σ)= \frac{ζ_{k(c^{1/3})}(σ)}{ζ_k(σ)}, \end{equation*} as $c$ varies. Moreover, we express the characteristic function of $F_{\sigma }$ explicitly as a product indexed by the prime ideals of ${\mathfrak{O}}_k$. 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 $\{k(c^{1/3}){\}}_c$. We also deduce a similar result for the Euler-Kronecker constants of this family.