Zeros of theta functions with characteristics ($z_0=(1/2-\alpha )\tau +(1/2-\beta )(\mathrm{mod}\mathbb{Z}+\mathbb{Z}\tau )$)

The logarithmic derivative is periodic under one and decreases by $2\pi i$ under tau. Integrating around a fundamental cell and applying the argument principle counts exactly one zero, with multiplicity. The displayed zero follows from translating the zero of $\theta [0,0]$ at $(1+\tau )/2$. It is consequently simple. For half-integer characteristics only the odd characteristic $(1/2,1/2)$ vanishes at the origin; the other three theta constants are nonzero.