Generalization of Riemann hypothesis for number fields

Statement

Suppose $K$ is a number field and ${\mathcal {O}}$ is the ring of integers in $K$. The hypothesis states the following: all the nontrivial zeros of the Dedekind zeta-function have real part $1/2$.

Here, the Dedekind zeta-function is defined as:

$\zeta _{K}(s):=\sum _{I}{\frac {1}{(N(I))^{s}}}$.

The sum is overall nonzero ideals of ${\mathcal {O}}$, and $N(I)$ is the index of $I$ in ${\mathcal {O}}$ as an additive subgroup.

This result is sometimes termed the generalized Riemann hypothesis or the extended Riemann hypothesis, but that name is typically used for the generalized Riemann hypothesis involving Dirichlet L-functions.