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The motivation to this question can be found in

What is the Hadamard's Factorization of a function that has a finite number of zeros

My question here is: I deduce that the function $h(s)$ has a finite number of roots in $(0,1)$. However how I can see that the corresponfing entire function $h(s)$ has also a finite number of roots in the whole complex plane.

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Without knowing at least something more about $h$, you can't conclude that. The function $f(z) = \sin z$ has no zeros in $(0,1)$ but infinitely many zeros in the whole plane.

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  • $\begingroup$ @ mrf: The function $f$ is the derivative of the Dirichlet series of the Hasse--Weil L-function of a modular curve $C$ over $Q$ . $\endgroup$ – China Jan 26 '13 at 10:02
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    $\begingroup$ Is the $f$ in this comment the $h$ in the question? and, if so, why was this information not in the question? (and, if not, then what does it mean?) $\endgroup$ – Gerry Myerson Jan 26 '13 at 11:30
  • $\begingroup$ @ Gerry Myerson : No, $f$ in this comment is not the $h$ in the question. $\endgroup$ – China Jan 26 '13 at 13:06

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