Singularities of symmetric powers and irrationality of motivic zeta functions

Let $K_0({\mathcal{V}}_K)$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L}=[{\mathbb{A}}_K^1]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational singularities, then all its symmetric powers also have $\mathbb{L}$-rational singularities. We then use this result to show that, for a smooth complex projective variety $X$ of dimension greater than one, the rationality of its Kapranov motivic zeta function $Z(X,t)$ (viewed as a formal power series over $K_0({\mathcal{V}}_{\mathbb{C}})$) implies that the Kodaira dimension of $X$ is negative and that $X$ does not admit global nonzero differential forms of even degree. This extends the irrationality part of the Larsen-Lunts rationality criterion from the surface case to arbitrary dimension. We also discuss some applications of these results.