Deligne–Beilinson rationality conjecture for motivic L-values

Let $M$ be a $K$-motive, let $\Delta_K(M)$ be its fundamental line, and let $\vartheta_\infty:K_{\mathbb R}\otimes_K\Delta_K(M)\cong\mathbf{1}_{K_{\mathbb R}}$ be the period-regulator isomorphism supplied by the preceding conjecture. For each embedding $K\to\mathbb C$, write $L_K^*(M):\mathbf{1}_{\mathbb C}\to\mathbf{1}_{\mathbb C}$ for the leading $L$-value viewed over $\mathbb C$. Deligne–Beilinson rationality conjecture. There is a unique isomorphism

$$\zeta_K(M):\mathbf{1}_K\to\Delta_K(M)$$

such that, after every embedding $K\to\mathbb C$, its composite with $(\vartheta_\infty)_\mathbb C$ is $L_K^*(M)$. This predicts that the leading value is rational after accounting for the fundamental line, periods, and regulators; the source explains that it connects the complex special value with the $p$-adic theory.