Let $$\theta(t) = \sum_{n \in \mathbb Z}e^{-n^2 \pi t}.$$ We can derive a functional equation for $\theta$ using Poisson summation formula: $$\theta(1/t) = \sqrt t \theta(t). $$
Riemann uses the above functional equation for $\theta$ to write $$\pi^{-s/2}\Gamma(s/2) \zeta(s) = \int_0^{\infty} x^{\frac {s}{2}-1} \frac{\theta(x)-1}{2}dx $$ and then proceeds to establish the functional equation: $$\xi(s) = \xi(1-s) $$ where $\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s).$
Now, my question is just out of curiosity: Can we deduce functional equation of $\theta$ from the functional equation of $\zeta?$ My approach would be to use inverse Mellin's transform. Are there other approaches?