I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.

The question is this: the Weil conjectures assert that $Z=\frac{P_1(t)...P_{2r-1}(t)}{P_0(t)...P_{2r}(t)}$ where the $P_i$'s are certain polynomials. (the assertion is of course stronger, but the rest of it is besides the point) What is the deep reason that these $P_i$'s alternate between numerator and denominator?

In Milne's notes about motives , (http://www.jmilne.org/math/xnotes/MOT.pdf), he asserts the following: let $hX$ be the motive corresponding to $X$, and let $h^0X$,...,$h^{2r}X$ be the conjectured decomposition into pure motives (conjecture C in Milne). He then says: define for a pure motive of weight $k$ the zeta function as the characteristic polynomial of the Frobenius if $k$ is odd, and its inverse if $k$ is even. Then extend the definition to motives by: the zeta function of a direct product of motives goes to the product of the zeta functions of the individual motives. Then indeed: $Z(X,s)=Z(hX,s)=Z(h^0X,s)...Z(h^{2r}X,s)$, where $Z(h^kX,s)=P_k(t)$ for $k$ odd and $\frac{1}{P_k(t)}$ for $k$ even.

So one can reduce this question to: why are we defining the zeta function of a motive of weight $k$ to be the characteristic polynomial of the Frobenius or its inverse depending on the parity?

I've been doing some light(?) reading on motives and the standard conjectures in an attempt to put various things that I tangentially know in perspective.

The question is this: the Weil conjectures assert that $Z=\frac{P_1(t)...P_{2r-1}(t)}{P_0(t)...P_{2r}(t)}$ where the $P_i$'s are certain polynomials. (the assertion is of course stronger, but the rest of it is besides the point) What is the deep reason that these $P_i$'s alternate between numerator and denominator?

In Milne's notes about motives, he asserts the following: let $hX$ be the motive corresponding to $X$, and let $h^0X$,...,$h^{2r}X$ be the conjectured decomposition into pure motives (conjecture C in Milne). He then says: define for a pure motive of weight $k$ the zeta function as the characteristic polynomial of the Frobenius if $k$ is odd, and its inverse if $k$ is even. Then extend the definition to motives by: the zeta function of a direct product of motives goes to the product of the zeta functions of the individual motives. Then indeed: $Z(X,s)=Z(hX,s)=Z(h^0X,s)...Z(h^{2r}X,s)$, where $Z(h^kX,s)=P_k(t)$ for $k$ odd and $\frac{1}{P_k(t)}$ for $k$ even.

So one can reduce this question to: why are we defining the zeta function of a motive of weight $k$ to be the characteristic polynomial of the Frobenius or its inverse depending on the parity?