Dedekind zeta function

Remarks

The series converges absolutely for $\operatorname{Re} s > 1$. Unique factorisation of ideals gives the Euler product $\zeta_K(s) = \prod_{\mathfrak{p}} \big(1 - N(\mathfrak{p})^{-s}\big)^{-1}$ over the nonzero prime ideals $\mathfrak{p}$. The function has an analytic continuation to a meromorphic function on $\C$ whose only pole is a simple pole at $s = 1$. Let $K$ have $r_1$ real embeddings, $r_2$ pairs of complex embeddings, and $w_K$ roots of unity. Let $h_K$ be its class number, $R_K$ its regulator and $d_K$ its discriminant. The residue is $$\operatorname{Res}_{s = 1} \zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} h_K R_K}{w_K \sqrt{\abs{d_K}}},$$ the class number formula.

Example: for $K = \Q$ every nonzero ideal is $n\Z$ for a unique $n \ge 1$, with norm $n$, so $\zeta_{\Q}$ is the Riemann zeta function. For $K = \Q(i)$, comparing Euler factors prime by prime gives $\zeta_K(s) = \zeta(s) L(s, \chi_4)$, where $L(s, \chi_4)$ is the Dirichlet L-function of the non-trivial character modulo $4$.

Non-example: the sum $\sum_{\alpha} \abs{N_{K/\Q}(\alpha)}^{-s}$ over nonzero elements $\alpha \in \mathcal{O}_K$ is not $\zeta_K(s)$. It counts each principal ideal once for each unit, so for $K = \Q(\sqrt{2})$, which has infinitely many units, it diverges for every $s$.