= Functional equation of an additive cusp-form twist
{title2=$M(f,a/N,k-s)=i^kM(f,-d/N,s)$}
For a level-one <cusp form> $f$ of even weight $k$, complete the <additive twist of a cusp-form L-function> as $M(f,a/N,s)=(N/(2\pi))^s\Gamma(s)L(f,a/N,s)$. Split its <Mellin transform> at $y=1/N$ and use <rational-cusp inversion of a level-one cusp form> in the lower integral. The resulting two integrals over $[1/N,\infty)$ converge locally uniformly for every complex $s$, giving an entire continuation and the displayed <functional equation>. The relation $ad\equiv1\pmod N$ identifies the opposite cusp.
Here $N$ is the positive denominator of a reduced rational <modular cusp>, not the level of $f$; take $\gcd(a,N)=1$ and $ad\equiv1\pmod N$.
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