Let $\chi$ be an odd Dirichlet character modulo $N$. Let $E_{1,\chi}\in M_1(N,\chi)$ be the (normalized) weight 1 Eisenstein series of $\chi$. If $\chi$ is primitive, this series has $q$-expansion \begin{equation}\tag{1} E_{1,\chi}(z)=\frac{ L(\chi,0)}{2} + \sum_{n=1}^\infty \left(\sum_{d|n}\chi(d)\right)q^n, \end{equation} where $q=e^{2\pi i z}$. In this case, we can see explicitly that $L$-function of $E_{1,\chi}$ factors into a product of classical $L$-functions, namely \begin{align*} L(E_{1,\chi},s) &= \sum_{n=1}^\infty \left(\sum_{d|n}\chi(d)\right) n^{-s}\\ &= \sum_{a=1}^\infty \sum_{b=1}^\infty \chi(a) (ab)^{-s}\\ &= \left(\sum_{a=1}^\infty \chi(a)a^{-s}\right) \left(\sum_{b=1}^\infty b^{-s}\right)\\ &= L(\chi,s)\zeta(s). \end{align*}
Now, when $\chi$ is not primitive, one can still consider the Eisenstein series $E_{1,\chi}$, however, the $q$-expansion is given by the more complicated formula \begin{equation}\tag{2} E_{1,\chi}(z)=-\frac{NL(\bar{\chi},1)}{2\pi i\,\tau(\bar{\chi}')}+\sum_{n=1}^\infty \left(\sum_{d|n}\sum_{r|(d,h)}\mu\Big(\frac{h}{r}\Big)\bar{\chi}'\Big(\frac{h}{r}\Big)\chi'\Big(\frac{d}{r}\Big) r \right)q^n, \end{equation} where $\chi'$ is the primitive character of $\chi$ modulo its conductor $N'$, and $h=N/N'$ (this formula is taken from Miyake's book titled "Modular Forms", Theorem 7.2.13).
My question is: What happens to the $L$-function of $E_{1,\chi}$ when $\chi$ is not primitive? Can we still factor $L(E_{1,\chi},s)$ in this case?
In all the literature I have come across, factorizations such as this are handled only for primitive characters. Nonetheless, it seems (to me at least) that it should still be possible to factor for general $\chi$. Perhaps we would miss a few Euler factors or introduce an explicit error, and that is ok.
I have tried to manipulate the summations inside the coefficients in (2) to make it work, but I failed. The $a_n$ in (2) are way more complicated to rearrange than the ones in (1). I may be missing something.
Also, if someone knows a reference for these kinds of factorizations when $\chi$ is not primitive, it would be deeply appreciated.