Dedekind zeta function

Let $K$ be a number field with ring of integers ${𝒪}_K$. Then the Dedekind zeta function of $K$ is the analytic continuation of the following series:

$${\zeta }_K(s)=\sum _{I\subset {𝒪}_K}{(N_{\mathbb{Q}}^K(I))}^{-s}$$

where $I$ ranges over non-zero ideals of ${𝒪}_K$, and $N_{\mathbb{Q}}^K(I)=|{𝒪}_K:I|$ is the norm of $I$.

This converges for $\mathrm{\Re }(s)>1$, and has a meromorphic continuation to the whole plane, with a simple pole at $s=1$, and no others.

The Dedekind zeta function has an Euler product expansion,

$${\zeta }_K(s)=\prod _{𝔭}\frac{1}{1-{(N_{\mathbb{Q}}^K(𝔭))}^{-s}}$$

where $𝔭$ ranges over prime ideals of ${𝒪}_K$. The Dedekind zeta function of $\mathbb{Q}$ is just the Riemann zeta function.