A number field analogue of Weil's theorem on congruent zeta functions

Let $K$ be a function field of one variable over a finite field $\mathbb{F}$. Weil’s celebrated theorem states that the congruent zeta function of $K\mathrm{/}\mathbb{F}$ is determined by the $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{F}}\mathrm{/}\mathbb{F})$-module structure of $X_{\bar{\mathbb{F}}K}(p){\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p$, and vise versa, where $p$ is a prime number different from the characteristic of $K$ and $X_{\bar{\mathbb{F}}K}(p)$ stands for the Galois group of the maximal unramified abelian $p$-extension over $\bar{\mathbb{F}}K$. In the present paper, I will give a number field analogue of the above mentioned theorem by considering the total cyclotomic extension, which we may regard as a number field analogue of $\bar{\mathbb{F}}K\mathrm{/}K$.