regarding a proof of $\|\theta(e^{i\lambda})\|^2$

Question

When studying the spectral representation of time series, I read the following formula,

I am not clear how to prove the second equation.

I expand the left side of the second equation with the $\theta(e^{i\lambda})$ defined as in the first equation, but how to derive its equivalence with its right side. In specific, I am kind of stuck with the involvement of $e^{-i\lambda}$.

Answer 1

$$\left| \theta(e^{i \lambda}) \right|^2=\theta(e^{i \lambda}) \overline{\theta(e^{i \lambda})}= \prod_{j=1}^q (e^{i \lambda}-z_j) \cdot \prod_{j=1}^q (e^{-i\lambda}-\bar{z_j})=\prod_{j=1}^q \left\{(e^{i \lambda}-z_j)(e^{-i \lambda}-\bar{z_j}) \right\}. $$