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 $\sigma >\frac{1}{2}$, we prove an asymptotic distribution function $F_{\sigma }$ for the values of the logarithm of

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

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