Value-distribution of quartic Hecke L-functions

Abstract

Set $K=\mathbb{Q}\left(i\right)$ and suppose that $c\in \mathbb{Z}\left[i\right]$ is a square-free algebraic integer with $c\equiv 1\left(mod⟨16⟩\right)$. Let $L\left(s,{\chi }_c\right)$ denote the Hecke $L$-function associated with the quartic residue character modulo $c$. For , we prove an asymptotic distribution function $F_{\sigma }$ for the values of the logarithm of

$$L_c\left(s\right)=L\left(s,{\chi }_c\right)L\left(s,{\overline{\chi }}_c\right)$$

as $c$ varies. Moreover, the characteristic function of $F_{\sigma }$ is expressed explicitly as a product over the prime ideals of $\mathbb{Z}\left[i\right]$.