Beilinson–Deligne rationality conjecture for the fundamental line

Let $\mathcal{M}$ be the modular motive, let $\Delta(\mathcal{M})$ be its one-dimensional fundamental line over $F$, and assume the non-degeneracy conjecture for the Gillet–Soulé height pairing so that there is an isomorphism $\theta_\infty:\Delta(\mathcal{M})_\infty\overset\simeq\longrightarrow F_\infty$. Let $L^*(\mathcal{M},0)$ be the leading term of the $L$-function at $s=0$. Rationality conjecture. There exists $\zeta_f\in\Delta(\mathcal{M})$ such that

$$\theta_\infty(\zeta_f)=L^*(\mathcal{M},0)^{-1}$$

in $F_\infty^\times$. This is a motivic rationality conjecture attributed in the source to Beilinson and Deligne; the paper proves it in analytic ranks $0$ and $1$ under technical conditions.