Take the 2-minute tour ×

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").

share|improve this question
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

1 Answer 1

up vote 2 down vote accepted

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,...

share|improve this answer
Thanks Stopple. –  joro Jan 14 '13 at 16:32

Your Answer


