Character-twisted Eisenstein series ($G_k(\chi ,z)$)

For $k>2$ and a Dirichlet character modulo $N$ with $\chi (-1)=(-1{)}^k$, define $G_k(\chi ,z)={\sum }_{m,n\in \mathbb{Z},(n,N)=1}\chi (n)(mz+n{)}^{-k}$. This series converges absolutely and locally uniformly on the complex upper half-plane and has period $N$.

With $S_{\chi }(r)$ the finite Fourier transform in Gauss sum of a Dirichlet character, the constant coefficient of $G_k(\chi ,z)$ in $e^{2\pi iz/N}$ is $2L(\chi ,k)$, and its positive coefficients areFor a primitive Dirichlet character this simplifies to $2(-2\pi i{)}^{k}g(\chi )((k-1)!N^k{)}^{-1}{\sum }_{r\mid n}\overline{\chi }(r)r^{k-1}$. Without primitivity the simplified formula is generally false.