Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/N^2\mathbb{Z})[x]$ where
$$f(x) := \prod_{k=0}^{N-2} (x-g^k)$$
I know that this polynomial in $\mathbb{Z}/N\mathbb{Z}[x]$ is equivalent to $(x^N-1)$ because they are both monic polynomials of the same degree with the same roots.
Similarly in $\mathbb{Z}/N^2\mathbb{Z}[x]$, $$\prod_{k=0}^{N(N-1)-1} (x-g^k)=(x^{N(N-1)}-1)$$
I have tried generating a few small examples of this product but have not seen an obvious pattern emerge. This polynomial also seems different from cyclomatic polynomials because the roots are close to each other rather than separated by by a gcd relation.
Ideally, I want to evaluate $f(x)$, but I am also curious what research has been done on these partial products. My preliminary searches have failed to find a name for this polynomial much less any results, so any information would be informative.
Note: This equation and question are very similar to the second quantity asked about in https://math.stackexchange.com/questions/3398653/polynomial-whose-roots-are-some-of-the-nth-roots-of-unity. But not much by way of follow up has been given for possible ways to evaluate or simplify the polynomial
