Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to $\mathrm{dim}_{\mathbb{Q}} (E(\mathbb{Q}) \otimes_{\mathbb{Z}} \mathbb{Q})$. There is a $c \in \mathbb{C}^\times$ such that $L(E, s) \sim c \cdot (s - 1)^r$ as $s \rightarrow 1$. Birch and Swinnerton-Dyer predict a great deal about $c$, in particular, that

- $c \in \mathbb{R}$,
- $c \in \mathbb{R}_{> 0}$,
- $c/\Omega_E \in \mathbb{Q}$ if $r = 0$, where $\Omega_E$ is the volume of $E(\mathbb{R})$ computed with respect to "the" Neron differential of $E$, and
- $c/\Omega_E \in \mathbb{Q}_{> 0}$ if $r = 0$.

Which of these predictions are known to hold?

Partial answers and answers in special cases (e.g., in the case when $E$ has potential complex multiplication) are very welcome!

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