Could a motivic spectrum have a "zeta function"?

Question

I'm currently learning about zeta functions, so I apologize in advance if this is riddled with nonsense. Suppose you have a sequence $E=(E_0,E_1,...)$ of motivic spaces along with structure maps $s_i:\mathbb{P}^1\wedge E_i\to E_{i+1}$ that induce weak equivalences $\Sigma_{\mathbb{P}^1}E_i\simeq E_{i+1}$, so that $E$ is a motivic spectrum. Is there a sequence of zeta functions $\zeta_E=(\zeta_{E_0},\zeta_{E_1},...)$ that respects the structure maps?

There are several pieces of this question that I'm not sure about:

1. If $E_i$ is a projective variety over $\mathbb{F}_q$, then we can assign a local zeta function $\zeta_{E_i}$ to it. Is it possible to do the same for a more general motivic space?
2. If $X$ and $Y$ are motivic spaces that have zeta functions $\zeta_X$ and $\zeta_Y$, is there a zeta function $\zeta_{X\wedge Y}$ for their smash product? If so, is there any way to express $\zeta_{X\wedge Y}$ in terms of $\zeta_X$ and $\zeta_Y$?
3. If $X$ and $Y$ are motivic spaces that have zeta functions $\zeta_X$ and $\zeta_Y$, and if $f:X\to Y$ is a sufficiently nice morphism, is there a way to make sense of $f_*\zeta_X$ (for example, would this be $\zeta_{f(X)}$)? If so, is there a way to relate $f_*\zeta_X$ and $\zeta_Y$?

Answer 1

Etale cohomology factors through the stable $\mathbb A^1$ homotopy type. So you should simply consider your local zeta factor in terms of traces of frobenius on etale cohomology. Since smashing with $\mathbb P^1$ corresponds to tensoring with $\mathbb Q_l(q)$ the eigenvalues that give you the local factor will be multiplied by $q$ each time. This means that the local factor will undergo the transformation $f(t) \mapsto f(qt)$. In terms of the global zeta function, this should correspond to translating by $1$.