Define the theta-functions $g(z) = - \sum_{n \in \mathbf{Z}} \left(n + \frac{1}{6} \right) e^{3 \pi i \left( n + \frac{1}{6} \right)^2 z}$. How can I show that $$g(z + 1) = e^{\frac{1}{24}2 \pi i} g(z)$$ and $$g\left(-\frac{1}{z} \right) = - (-i z)^{\frac{3}{2}} \sum_{n \in \mathbf{Z}} (-1)^n \left(n + \frac{1}{3} \right) e^{3 \pi i \left( n + \frac{1}{3} \right)^2 z}$$ This example is from S. Zweger's paper "Mock Theta Functions and Real Analytic Modular Forms", where he states that this can be shown "using standard-methods". I don't know those standard-methods, so I'd appreciate some help.
Thanks!