The generic nonvanishing conjecture for definite Heegner-family $L$-values

Let $f_\mathfrak p^\dagger$ be the twisted specialization at an arithmetic prime $\mathfrak p$ of weight $k_\mathfrak p\ge2$, let $\chi$ be a fixed character of $\widetilde G_c$ as in the source, and let $L_K(f_\mathfrak p^\dagger,\chi,s)$ be the corresponding $L$-function. Let $\mathcal L(f_\infty/K,\chi,\mathfrak p)$ be the specialized algebraic $L$-value element. The generic nonvanishing conjecture. Assume that $w=1$. Then

$$L_K\bigl(f_\mathfrak p^\dagger,\chi,k_\mathfrak p/2\bigr)\ne0 \quad\Longleftrightarrow\quad \mathcal L(f_\infty/K,\chi,\mathfrak p)\ne0.$$

Equivalently, the critical $L$-value vanishes precisely when the corresponding family element lies in $\mathfrak p$. The conjecture refines generic analytic-rank predictions and is related in the source to Bloch–Kato Selmer-group predictions.