The odd-weight invertible motive conjecture

Let $k$ be a field, let $\mathcal{M}(k)$ be the category of motives over $k$ for a fixed adequate equivalence relation, and let $L\in\mathcal{M}(k)$ be an invertible motive. Define the weight of $L$ as the weight of its realisation under a suitable classical Weil cohomology theory. Odd-weight invertible motive conjecture. There is no invertible motive $L$ of odd weight $n$ in $\mathcal{M}(k)$. The conjecture concerns the possible weights of invertible motives and is used in the paper to derive parity properties of Euler characteristics and simplify the functional equation for motivic zeta functions; its general resolution is not stated.