Well, Peter's answer is overkill for this particular problem. While this zeta-function will certainly be a Burgess zeta-function, the study of the zeta-function of this particular kind will be much simpler, and its properties can be directly deduced from properties for the Dirichlet L-functions.  For simplicity I will show how to do this in the case $\chi(n)=1$ in your question, although the general character case can be treated similarly, since if we assume that $\chi$ is a character mod $N$ then $\chi \chi_1$ will be a character mod $Nq$ whenever $\chi_1$ is a character mod $q$. 

Let
$$ B(s)=\prod_{p \equiv a \pmod q} (1-p^{-s})^{-1}.$$
Taking the logarithm we find that
$$\log B(s)=  \sum_{n=1}^\infty \frac{B_0(ns)} n,$$
where
$$ B_0(s)= \sum_{p \equiv a \pmod q} p^{-s}$$
is some variant of the prime zeta-function. For the half plane Re$(s)>1/2$ the terms when $n \geq 2$ will be absolutely convergent and the main term will be $B_0(s)$.   For the Dirichlet $L$-series $L(s,\chi)$ we have similarly that
$$ \log L(s,\chi) =  \sum_{n=1}^\infty \frac{L_0(ns,\chi^n)} n,$$
where
$$  L_0(s,\chi)= \sum_{p} \chi(p) p^{-s}.$$ 
By Möbius inversion we get 
$$L_0(s,\chi)= \sum_{n=1}^\infty \frac{\mu(n)}{n} \log L(ns,\chi^n).$$
It is simple to see from the definitions of the Dirichlet series and using the fact that $\sum_{\chi \pmod q}\chi(a)=\phi(q)$ if $a \equiv 1 \pmod q$ and 0 otherwise
that
$$ B_0(s)=  \frac 1 {\phi(q)} \sum_{\chi \pmod q} \overline{\chi(a)} L_0(s,\chi).$$ 
By combining these results we find that 
$$  \log B(s)= \frac 1 {\phi(q)} \sum_{\chi \pmod q} \overline{\chi(a)} \sum_{n=1}^\infty \frac 1 n \sum_{d|n} \mu \left(\frac n d \right) \log L(ns,\chi^d). 
$$


 The most important term will come from $n=1$ since the other terms will be absolutely convergent for Re$(s)>1/2$. Thus we have that
$$ \log B(s)=\frac 1 {\phi(q)} \sum_{\chi \pmod q} \overline{\chi(a)}\log L(s,\chi)+ R(s),$$ where $R(s)$ is absolutely convergent for Re$(s)>1/2$. This means that we can write
$$ B(s)= \prod_{\chi \pmod q} L(s,\chi)^{\overline{\chi(a)}/\phi(q)} A(s),$$
where $A(s)$ is a Dirichlet series absolutely convergent and nonvanishing for Re$(s)>1/2$. In particular it means that under the Generalized Riemann hypothesis $(s-1)B(s)^{\phi(q)}$ will be a holomorphic nonvanishing function for Re$(s)>1/2$. By this method it will be possible to get an analytic continuation up to Re$(s)=0$ (its natural boundary should be Re$(s)=0$ since singularities coming from the zeros of the L-functions will be dense close to that line), with exeption for singularities at $\rho/n$ where $\rho$ is a zero of some Dirichlet L-function and $1/n$.  

Thus the study of the analytic properties of this zeta-function will be simple consequences of the properties of the Dirichlet L-functions.