Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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
1  
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
add comment

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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.