I'm trying to find the positions or at least the number of zeros of $\partial_z\vartheta[p,0](z,\tau)$ where the standard theta function with characteristics $(p,q)$ is defined as $\vartheta[p,q](z,\tau) = \sum_{m\in\mathbb{Z}}e^{\pi i (m+p)^2\tau + 2\pi i (m+p)(q+z)}$
In particular, any information about the cases where $p,z$ are restricted to real numbers would be great. Thank you guys in advance :)