Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.

Assume $F_q$ to be a finite field with q elements. Consider the zeta function of the hypersurface defined by $-y_0^2+y_1^2+y_2^2+y_3^2=0$ in $\mathbb{P}^3$.

If $-1$ is a square in $F_q$, the zeta function is

$$Z(u)=\frac{1}{(1-uq^2)(1-uq)^2(1-u)}.$$

It has a pole of order $2$ at $1/q$. If not, it's

$$Z(u)=\frac{1}{(1-uq^2)(1-uq)(1+uq)(1-u)}.$$

It has a pole of order $1$ at $1/q$.

How does orders of poles indicate any geometric information?