Dedekind zeta function

Summary

Let $K$ be an algebraic number field. Its Dedekind zeta function $\zeta_K(s)$ is the sum of $N(\mathfrak{a})^{-s}$ over the nonzero ideals $\mathfrak{a}$ of the ring of integers. For $K = \Q$ it is the Riemann zeta function. It has an Euler product over the prime ideals and a functional equation. It extends to a meromorphic function on the whole complex plane with a single simple pole at $s = 1$. Its residue at $s = 1$ is given by the class number formula. This formula ties together the class number, the regulator and the discriminant.