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 $$B(s)=\sum_{p\equiv a\pmod{q}} \left(1-\frac{\chi(p)}{p^s}\right)^{-1}.$$

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.