The Riemann $\Theta$-function is defined as $$Θ(z|Ω)=\sum\limits_{q∈Z^N}e^{πiq⋅Ωq+2πiq⋅z},$$ where $\Omega$ has positive definite imaginary part to ensure convergence. In a particle physics calculation I encountered this series, except $\Omega$ is purely imaginary and positive semi-definite. In particular I have $N=2k$, and $q\cdot\Omega q=0$, iff $q=(v,v)$ with $v\in Z^k$.
Clearly $\Theta(z\vert\Omega)$ does not converge in this case, so my question is: Is it possible to analytically continue $\Theta(z\vert \Omega)$ to $\Omega$'s with positive semidefinite imaginary parts? Or if this is too hard: Does anyone have an idea about analytically continuing the particular case above?
Thanks!