Analytic class number formula (reviewed)

If $K$ is a number field with signature $(r_1, r_2)$, discriminant $D$, regulator $R$, class number $h$, containing $w$ roots of unity, and Dedekind $\zeta$-function $\zeta_K$, then $\zeta_K$ has a meromorphic continuation to the whole complex plane with a single pole at $s=1$, which is of order $1$. The analytic class number formula gives the residue at this pole: $$ \lim_{s\to 1}\ (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} .$$