The Density

The following conjecture is not mentioned elsewhere in this paper.

Conjecture 4.2   Let $p$ be a rigid prime for an elliptic curve $E$. The set of primes

$$\left\{\ell : \ell \equiv 1\!\!\!\!\!\pmod{p}\text{ and } L(E,\chi_{p,\ell},1)=0\right\}$$

has Dirichlet density 0.

The following numerical example gives evidence for this conjecture.

Example 4.3   Let $E$ be 37A and let $p=5$. Then the only $\ell<1000$ (with $\ell\equiv 1\pmod{5}$) for which $L(E,\chi_{5,\ell},1)=0$ is $\ell=41$.