# nt.number theory – Is there a Siegel-Weil formula for generalized Siegel theta functions?

For the usual Siegel-Narain theta functions $$Theta(m,tau)$$, the Siegel-Weil formula relates the integral of it on the 2D dimensional Narain moduli space to the Eisenstein Series on $$SL(2,Z)$$ as $$E_{D/2}(tau)$$.
To be more concrete

$$int _{{mathcal{M}}} dmu Theta(m,tau)=E_{D/2}(tau)$$

where m denotes a point in the moduli space.

I wonder if there is an analogous formula for the generalized Siegel theta functions $$Theta (alpha,beta)(m,tau)$$?

where $$Theta (alpha,beta)(m,tau)$$ means that we do some shifts in the lattice sum by vectors related to $$alpha$$ and $$beta$$. (twisted partition function for the compact boson in physics language)