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.