I have a question: I always saw the functional equation for a $L(s,\chi)$-function, and every time it is specified that the character $ \chi \bmod q $ is primitive. In my understanding the problem is that in the proof (that I saw) is that we used the fact that $$ \chi(n) = \frac{G(\overline{\chi},n)}{G(\overline{\chi},1)} $$ where $G(\chi,n)$ is the Gauss sum. This is equivalent to saying that the character is primitive.
My question: is it possible to derive a similar functional equation if $\chi$ is not primitive?
Let $ \chi $ be a primitive Dirichlet character of modulo $q>1$ and $ k \in \{0,1\} $ defined such that $ \chi(-1) = (-1)^k$, denote with $ G(\chi,n)$ the Gauss sum. Then the functional equation is $$ \Lambda(s,\chi) = \frac{G(\chi,1) }{i^k \sqrt{q}} \Lambda(1-s,\overline{\chi}) $$ Where $ \Lambda(s,\chi)$ is the completed $L$-function, defined as $$ \Lambda(s,\chi) = \left( \frac{q}{\pi} \right)^{\frac{s+k}{2}} \Gamma \left( \frac{s+k}{2} \right)L(s,\chi) $$