I consider the equation $x^2+y^2+z^3=xyz+2$ over the integers. As a variety this only has a singularity in characteristic $2$. Now I am interested in calculating the arithmetic zeta function of this variety. It is doable to calculate the Hasse-Weil zeta function, which already gives all factors except for characteristic $2$ reduction.
The first problem is that I am not sure how the factor is even defined, as almost all sources just talk about the good reduction case.
What I hoped to do was to use the definition as $$ \zeta_X(s) = \prod_{x}1/(1-N(x)^{-s}). $$ If I understand correctly, to obtain the $p=2$ factor we take a product over all powers of $2$ and get a factor $1/(1-q^{-s})^{q^2+1}$ for $q=2^n$ (here I use that there are $q^2+1$ solutions in the field of $q$ elements). But it does not completely feel right to me.
Another option is using the characteristic polynomial of Frobenius at p on the inertia-invariant subspace of the $\ell$-adic cohomology of the variety. But for this option I have no idea how to even start computing any of this.
At last an option would be to magically find an non-singular model over $\mathbb{Z}$, but with some basic operations I cannot find such a model. Maybe it would also be a bit easier to look at the projective closure over the equation, for the Hasse-Weil part that would not be a problem. So if that would make this bad-reduction factor easier it might also be a good idea.