  The zeta function of a motive over a finite field is multiplicative with
respect to the direct sum of motives. It has beautiful analytic properties, as
were predicted by the Weil conjectures. There is also a multiplicative zeta
function, which instead respects the tensor product of motives. There is no
analogue of the Weil conjectures, and we give a sufficient criterion for an
analytic continuation to exist. This happens, for example, for cellular
varieties, abelian varieties, or genus g > 1 curves with a supersingular
Jacobian.
