= Zeros of theta functions with characteristics
{title2=$z_0=(1/2-\alpha)\tau+(1/2-\beta)\pmod{\mathbb Z+\mathbb Z\tau}$}
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.
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