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 $1/q$.



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> How does orders of poles indicate any geometric information?
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How does the order of a pole of a zeta function indicate any geometric information?
For a smooth projective surface, the order of the pole at 1/q is conjectured to be the rank of the Neron-Severi group of the surface. That's a conjecture of Tate and is an analog of the Birch and Swinnerton-Dyer conjecture. Tate has formulated a more general conjecture for higher dimensional varieties too. For the case of quadrics, as in your example, these conjectures are known.


Edit: maybe you don't want a fancy answer. In the first case, the quadric contains lines and in the second, it doesn't.


