Riemann Siegel theta function
From specialfunctionswiki
The Riemann Siegel $\vartheta$ function is defined for $t \in \mathbb{R}$ by $$\vartheta(t) = \mathrm{Im} \left( \log \Gamma \left(\dfrac{1}{4}+ i \dfrac{t}{2} \right) - \dfrac{t \log \pi}{2} \right),$$ where $\log$ denotes the logarithm and $\Gamma$ denotes the gamma function.
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Graph of $\vartheta$ on $[-30,30]$.
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Domain coloring of analytic continuation of $\vartheta$.
