Holomorphic Eisenstein series (awaiting review)

Let $k, N_1, N_2$ be positive integers, and let $\chi_1, \chi_2$ be primitive Dirichlet characters modulo $N_1$ and $N_2$ respectively.

The Eisenstein series of weight $k$ associated to $\chi_1$ and $\chi_2$ is $$ E_{k}^{\chi_1, \chi_2}(\tau) = \frac{1}{2} \left ( \delta_{\chi_1=1} L(1-k, \chi_2) + \delta_{k=1} \delta_{\chi_2=1} L(0,\chi_1) \right) + \sum_{n=1}^{\infty} \sigma_{k-1}^{\chi_1, \chi_2}(n) q^n, $$
where $q = e^{2 \pi i \tau}$, $L(s,\chi_i)$ is the Dirichlet $L$-function associated to $\chi_i$, and $$ \sigma_{k-1}^{\chi_1, \chi_2}(n) = \sum_{m \mid n} \chi_1(n/m) \chi_2(m) m^{k-1}. $$