Dedekind zeta function

For a number field $K$, the Dedekind zeta function is formally

Unique factorization of nonzero ideals and multiplicativity of the ideal norm give the formal identitywhere the product ranges over nonzero prime ideals.

For a quadratic field $K$ of discriminant $D$,where ${\chi }_D(p)$ is $1$, $-1$, or $0$ according as $(p)$ splits, remains prime, or ramifies in ${\mathcal{O}}_K$. Comparing the three possible local Euler factors proves the formal identity.

For a number field of signature $(r_1,r_2)$, roots-of-unity count $w_K$, regulator $R_K$, class number $h_K$, and discriminant $d_K$,

For an imaginary quadratic field of fundamental discriminant $D<-4$,Equivalent finite character-sum forms follow by evaluating the odd quadratic Dirichlet L-function at one.