I have to compute the following quantity
\begin{align} \prod_{\alpha\neq\beta=0}^{N-1} \left(1-e^{2\pi i \frac{\alpha-\beta}{aN}}\right)\label{a}\tag{1} \end{align}
where $a$ is a natural number, or alternatively the quantity
\begin{equation} \prod_{\alpha=0}^{N-1} \left(x-e^{2\pi i \frac{\alpha}{aN}}\right)\label{b}\tag{2} \end{equation}
that is in other words the polynomial whose roots are the first $N$ among the $(aN)^{th}$-roots of unity.
I would like to know if there is some closed form expression for these quantities. The thing I know is that if I run the product over the whole set of the $(aN)^{th}$-roots of unity I have
\begin{equation} \prod_{\alpha=0}^{aN-1} \left(x-e^{2\pi i \frac{\alpha}{aN}}\right) = x^{aN}-1\label{c}\tag{3} \end{equation}
for the second quantity and accordingly
\begin{gather} \prod_{\alpha\neq\beta=0}^{aN-1} \left(1-e^{2\pi i \frac{\alpha-\beta}{aN}}\right) = \left.\frac{\prod_{\alpha,\beta=0}^{aN-1} \left(x-e^{2\pi i \frac{\alpha-\beta}{aN}}\right)}{(x-1)^{aN}}\right|_{x=1} = \left.\left(\frac{(x^{aN}-1)}{x-1}\right)^{aN}\right|_{x=1} = (aN)^{aN}\label{d}\tag{4} \end{gather}
for the first one. Thanks in advance for any help!
$$\left\|\prod_{k=0}^{N-1}\left(1-\exp\frac{2\pi i k}{a N}\right)\right\|=2^N\prod_{k=0}^{N-1}\sin\frac{\pi k}{a N}=2^N\exp\sum_{k=0}^{N-1}\log\sin\left(\frac{\pi}{a}\cdot\frac{k}{N}\right) $$ and by Riemann sums $$ \frac{1}{N}\sum_{k=0}^{N-1}\log\sin\left(\frac{\pi}{a}\cdot\frac{k}{N}\right)\approx \int_{0}^{1}\log\sin\left(\frac{\pi x}{a}\right)\,dx=\log\frac{\pi}{a}+O\left(\frac{1}{a^2}\right) $$ so $$ \log\left\|\prod_{k=0}^{N-1}\left(1-\exp\frac{2\pi i k}{a N}\right)\right\|\ll N\log\left(\frac{2\pi}{a}\right). $$