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Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the complete $L$-function (In the sense of Iwaniec's book Analytic number theory page 94) of the twisted $L$-function of $f$ by a caracter $\psi\; :$ $$L(\psi\otimes f,s)=\sum_{n\ge1}\frac{a(n)}{n^s}\psi(n)$$ and his functional equation. Can someone give me a reference in which i can find it ?

Thanks!

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    @user1952009, Thanks for your comment. – 2017-01-04
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    You know [this method](http://www.les-mathematiques.net/phorum/read.php?5,1308891,1344176#msg-1344176) for defining $L(f \times g,s)$ when $f,g \in M_k(\Gamma_0(N)),S_m(\Gamma_0(M))$ ? In the case you wrote ($k$ half-integral weight and $\chi$ non-trivial) the Poisson summation formula and the real Eisenstein series become messy, and the [adelic formulation](http://www.math.umn.edu/~garrett/m/mfms/) becomes a good idea. – 2017-01-04
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    The answer is reasonably straightforward (and the proof follows by classical means) when the conductors of $f$ and $\psi$ are coprime. See Proposition 14.20 of Iwaniec and Kowalski. When the conductors of $f$ and $\psi$ are not coprime, however, then you really have no recourse but to work adelically. – 2017-01-05
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    ...well, that's not quite true, because in some cases (e.g. when the conductor of $f$ is squarefree) you can still do it classically. But it becomes quite complicated. – 2017-01-05

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