If $f$ and $g$ are primitive modular forms of characters $\chi$ and $\psi$, such that $\chi, \psi$ and $\chi * \psi$ are all primitive, then we have an explicit functional equation. This is proven in Hida's book Elementary theory of L functions and Eisenstein series, section 9.5. In this book, he says that it is possible to remove the assumption about the primitivity of the characters, either by using harmonic analysis, or with a "large amount of technicality". (book available here)
As I don't know harmonic analysis, I tried to prove the functional equation without the primitivity assumptions by adding the large amount of technicality to the proof he gives in his article from 1985 "A p-adic measure attacher to the zeta functions associated with two elliptic modular forms I" (it seems to be essentially the same proof as in the book but with objects I'm more familiar with). I checked all the references he gives during the proof (especially the one from 1977 by Shimura "On the periods of modular forms") to find my way around but I've not yet succeeded : I split the Eisenstein series $F_{\alpha,N}$ using the article of Shimura, used the functional equation for the Eisenstein series, and I'm stuck just after the change of variables, as I can't seem to rebuild the series $F_{\alpha,N}$ from the pieces I have.
My question is thus : can anyone who knows how to do it give me some advice? Am I on the right track or should I do something different? Does there exist a reference for this explicit functional equation?
When Hida refers to "harmonic analysis" here, what he means (I think) is the theory of smooth representations of p-adic groups.
The definitive general treatment of this is Jacquet's 1972 Springer Lecture Notes volume #278. For a more down-to-earth account in the context of holomorphic modular forms, with applications to p-adic L-functions extending the ones given by Hida, there's a nice article recently by Chen and Hsieh, http://www.math.keio.ac.jp/~kurihara/11.ASPMstyle.pdf.
My impression is that if you try to do this "by hand" without invoking representation theory, then either you have to make very restrictive assumptions on the modular forms as Hida does, or you end up in a horrible baffling mess of cases which is impossible to navigate -- the representation theory exists precisely in order to give some structure to this mess.