Additive twist of a cusp-form L-function ($L(f,a/N,s)={\sum }_{n\ge 1}{c}_n{e}^{2\pi ian/N}{n}^{-s}$)

For a cusp form $f=\sum c_n{q}^n$ and coprime integers $a,N$, the additive twist inserts the additive character $e^{2\pi ian/N}$. It is the Dirichlet series associated with the rational translate $f(a/N+\tau )$. Its completed Mellin transform uses the scale $N^s$ and the Gamma function.

For 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 $ad\equiv 1(\mathrm{mod}N)$ identifies the opposite cusp.

If $ad\equiv 1(\mathrm{mod}N)$, the determinant-one matrix with rows $(a,(ad-1)/N)$ and $(N,d)$ sends $\tau -d/N$ to $a/N-1/(N^2\tau )$. The transformation of the cusp form gives the displayed identity. It turns the small-height portion of an additive-twist Mellin transform into exponentially decaying values at the inverse rational cusp.