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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 $$\zeta_{K, S}(s) : = \prod_{p \in S} \frac{1}{1-p^{-s}}.$$ More generally, one can define a similar partial Euler product for any splitting type, and ask the same question about the analytic properties of the resulting function. Any advice would be greatly appreciated.

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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 $$\zeta_{K, S}(s) : = \prod_{p \in S} \frac{1}{1-p^{-s}}.$$ More generally, one can define a similar partial Euler product for any splitting type, and the same question about the analytic properties of the resulting function. Any advice would be greatly appreciated.
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