Timeline for roots of analytic functions

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Oct 16, 2009 at 18:22	comment	added	Ben Webster♦		OK, that was something that should have gone in an edit of the question, not a comment.
Oct 16, 2009 at 17:30	comment	added	shenghao		Thanks. Let me give a second try, by imposing more requirements on f(z). I'm thinking about the case where f(z) is the zeta function of an algebraic stack over a finite field, so let's assume f(z) has an infinite product expansion over \ell-adic numbers, like \prod P_{odd}(z)/\prod P_{even}(z), where each P_i(z) is a polynomial over Q_{\ell} with constant term 1. Assume they have distinct weights, e.g. reciprocal roots of P_i(z) have weights i. Then can we conclude that all coefficients of P_i(z) are rational numbers? Thanks.
Oct 16, 2009 at 17:04	history	answered	Andy Putman	CC BY-SA 2.5