# Are potential complex zeros not on the critical line of Dedekind zeta function in quadruples?

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").

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Siegel zeros are real by definition... –  David Hansen Jan 13 '13 at 8:20
@David thank you, edited the question for complex zeros not on the critical line... –  joro Jan 13 '13 at 9:03
The functional equation tells you that if $\rho$ is a zero, so is $1-\rho$. On the other hand, since the Dirichlet series coefficients are real, we have that for $\text{Re}(s)>1$, $\zeta(\bar{s})=\overline{\zeta(s)}$. By analytic continuation this holds for all $s\ne 1$. So if $\rho$ is a complex zero off the critical line, so is $\bar{\rho}$. This works for any Dirichlet series with functional equation and real coefficients, i.e. real quadratic character, elliptic curve,...