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. 



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. 


