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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: $$ \lim_{s\to 1}\ (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} .$$

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2022-07-14 13:05:18
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