Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a Frobenius element and $M_\ell^I$ is the $\ell$-adic realization of $M$ stable under inertia $I_p$. On the other hand, the zeta function of $M$ is the infinite series $$Z(M,t)=\sum_{n=0}^\infty[Sym^nM]t^n$$ taking values in the ring of pure motives over $k$.

Is there a precise relation between the L-function and zeta function? Both seem to generalize the Hasse-Weil zeta function in different ways, but it is tempting (if nonsensical) to ask for an euler product for the series and vice versa. [EDIT: I believe this 'product' should in fact be the $\det(\dots)^{-1}$ factor, and then the global L-function defined as a product over almost all $p$]

Indeed, Dhillon and Minac (1991) define a 'motivic Artin L-function' as $L(M,\rho,t)=Z((V\otimes M)^G,t)$ where $\rho$ is a representation of a finite group $G$ in a $\mathbb Q$ vector space $V$, but it is not obvious whether $L(M,\rho,t)$ and $L(M,s)$ are directly related in any way.

The idea is to rewrite the $L$-function as the sum of $\sharp [Sym^nM](F_p)t^n$; here "the number of $F_p$-points" of a motif is a natural homomorphism from the Grothendieck group of motives to abelian groups that extends the "usual" number of points over the field $F_p$ for varieties. So, the $L$-function can be obtained from the motivic zeta via the application of this homomorphism. Possibly, Kapranov explians this somewhere; I have read about this in a survey on motivic integration.
• Mikhail — Can you provide a reference for this extension of the “number of $\mathbb{F}_{p}$-point”? Thanks. – jmc Jul 28 '14 at 6:13
• I did not read this 1991 paper; I can only say that you have just written the zeta function of the tensor product of $M$ by the Artin motif corresponding to $\rho$. The latter motif becomes constant when you pass to the field $F_{p^{\sharp G}}$, whereas $M$ can be an arbitrary motif. And here is a certain reference: users.ictp.it/~pub_off/lectures/lns019/Loeser/Loeser.pdf – Mikhail Bondarko Jul 30 '14 at 7:16