Deligne–Beilinson conjecture on the order of vanishing of motivic L-functions

Let $M$ be a $K$-motive, let $M^*(1)$ denote its Kummer dual, and let $\mathrm{H}^i_f(M^*(1))$ be its finite motivic cohomology groups. The order of vanishing $r(M)$ of $L_K(M,s)$ at $s=0$ is defined by

$$L_K(M,s)=L_K^*(M)s^{r(M)}+\ldots.$$

Deligne–Beilinson conjecture.

$$r(M)={\mathrm{dim}}_K \mathrm{H}^1_f(M^*(1))-{\mathrm{dim}}_K \mathrm{H}^0_f(M^*(1)).$$

This expresses the analytic order of vanishing in terms of motivic cohomology; the source presents it as part of the conjectural framework for special values of motivic $L$-functions.