This question arose from sums over zeros of Dedekind zeta function.

It is known that complex zeros of Dedekind zeta function are in pairs $\rho, 1 - \rho$.

Is it true that potential complex zeros not on the critical line of Dedekind zeta function must be in quadruples $\rho, 1 - \rho, \overline{\rho}, \overline{1 - \rho}$ ?

I am interested in the general case, not for specific number fields (or for number fields for which the answer is "no").