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A question about partial Euler products

Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of $$ \zeta_{K, S}(s) : = \prod_{p \i …
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A question about partial Euler products

This question turned out to be not too difficult. Please see http://www.math.uic.edu/~rtakloo/euler-product.pdf for a (casual) writeup of an answer. Thanks for your comments and hints.
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