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Let $F$ and $G$ be two functions belonging in the Selberg class, $\Lambda_{F}$ and $\Lambda_{G}$ the complete L-functions associated to $F$ and $G$. I would like to know whether this assertion is true or not:

"$\Lambda_{F}$ and $\Lambda_{G}$ have the same zeroes if and only if $F=G$ or $F=\overline{G}$."

Thank you in advance.

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If $F$ is in the Selberg class, so is $F^2$, $F^3$, etc. So, I think you need to assume $F$ and $G$ are primitive to ask this question. – Micah Milinovich Jan 18 '11 at 22:02
When I write "the same zeroes", I mean "the same zeroes with the same multiplicity for $\lambda_{F}$ and $\Lambda_{G}$", so that I don't think it's necessary to assume $F$ and $G$ are primitive. – Sylvain JULIEN Jan 18 '11 at 22:15
Maybe this is a naive comment, but why doesn't your conclusion follow from the Explicit Formula? – Stopple Jan 18 '11 at 22:52
Major typo: you write F=G or F=G !! This can't be what you intended; please edit! – Zen Harper Jan 19 '11 at 5:02
Zen, I see no typo: can't you see the overline on $G$ ? – Sylvain JULIEN Jan 19 '11 at 9:46
up vote 7 down vote accepted

Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question.

Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book Problems in Analytic Number Theory: Let $F$ be in the Selberg class, $n>1$ be a positive integer, and $T>1$. Then $$ \sum_{|\gamma|\leq T} n^\rho = -\frac{T}{\pi}\Lambda_F(n) + O( n^{3/2}\log T )$$ where $\rho=\beta+i\gamma$, $\beta>0$, runs over the non-trivial zeros of $F(s)$. Here the coefficients $\Lambda_F(n)$ are defined by $$ -\frac{F'}{F}(s) = \sum_{n=1} \frac{\Lambda_F(n)}{n^s}.$$

Now suppose that $F$ and $G$ are in the Selberg class and have the same zeros (with multiplicity). Then we deduce from Landau's formula that $$ |\Lambda_F(n) - \Lambda_G(n)| \ll \frac{n^{3/2}\log T}{T} $$ for all $n>1$. Letting $T\rightarrow \infty$, it follows that $\Lambda_F(n) = \Lambda_G(n)$ for all $n>1$. This implies that $F=G$.

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Thank you very much for this wonderful answer. – Sylvain JULIEN Jan 19 '11 at 21:14
By the way, has such a generalization of Landau's explicit formula been proven for automorphic L-functions? – Sylvain JULIEN Jul 29 '11 at 19:21

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