There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series
$$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$
or as an Euler product
$$\prod_{p\mbox{ prime}} \left(1-\frac{\chi(p)}{p^s}\right)^{-1}.$$

Correspondingly, this gives two ways of restricting a Dirichlet L-function to an arithmetic progression, by considering either
$$A(s)=\sum_{n\equiv a\pmod{q}} \frac{\chi(n)}{n^s}$$
or
$$B(s)=\prod_{p\equiv a\pmod{q}} \left(1-\frac{\chi(p)}{p^s}\right)^{-1}.$$

I am assuming here that *a* and *q* are relatively prime.  The function *A(s)* is a fairly classical object and has been studied extensively.

I am interested in finding out some analytic information on the function *B(s)*: can it be meromorphically continued to the complex plane?  What are its poles, if any?  What are the singular parts corresponding to these poles?  If that makes things any easier, I would even be happy to know about this in the special cases (*a*, *q*)=(1, 3) and (*a*, *q*)=(2, 3).

I have had no success in tracking this down, so I am hoping that somebody will either know of some references where this is worked out, or some hints on how I could go about doing it myself.