Let $K$ be a number field with ring of integers $\mc{O}_K$ . Then the Dedekind zeta function of $K$ is the analytic continuation of the following series: $$\zeta_K(s)=\sum_{I\subset\mc{O}_K} (N^K_{\Q}(I))^{-s}$$ where $I$ ranges over non-zero ideals of $\mc{O}_K$ , and $N^K_{\Q}(I)=|\O_K:I|$ is the norm of $I$ .

This converges for $\Re(s)>1$ , 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, $$\zeta_K(s)= \prod_{\fr p}\frac{1}{1-(N^K_{\Q}(\fr p))^{-s}}$$ where $\fr p$ ranges over prime ideals of $\mc{O}_K$ . The Dedekind zeta function of $\Q$ is just the Riemann zeta function.