Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 2 / 2 / 5 / Solution

Suppose $d_K<-4$. Then $K$ has signature $(0,1)$, regulator $1$, and exactly $w_K=2$ roots of unity. The analytic class number formula becomesBecause ${\zeta }_K(s)=\zeta (s)L(s,{\chi }_{d_K})$ and ${\mathrm{res}}_{s=1}\zeta (s)=1$,Putting $m=|d_K|$ into the supplied odd-character identity yields the quadratic class number formula