 Character-twisted Eisenstein series

For an integer $k>2$ and a Dirichlet character $\chi$ modulo $N\ge 1$ with $\chi (-1)=(-1{)}^k$, defineThe zero pair is explicitly excluded, including when $N=1$. Comparison with a convergent lattice Eisenstein sum proves absolute and locally uniform convergence on the complex upper half-plane, hence holomorphy. Replacing $z$ by $z+N$ permutes the terms by $n\mapsto n+mN$ without changing $\chi (n)$ or its coprimality condition, so the period is $N$. The parity condition makes the terms indexed by $(m,n)$ and $(-m,-n)$ agree instead of cancel.