*Edit due to the comment.*

Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable.

Given a sequence of irreducible unitary representations $(\pi_n)$ of $G(F)$ and a unitary representation $\pi$, are the following statements equivalent?

$\pi_n$ converges to $\pi$ in the Fell topology, i.e., for any matrix coefficient $f$ of $\pi$, there exists a sequence of matrix coefficients $f_n$ of $\pi_n$ with $f_n \rightarrow f$ locally uniformly.

The tuple $[ L(s, \pi_n, \psi), \epsilon(s, \pi_n, \psi)]$ converge to $[ L(s, \pi, \psi), \epsilon(s, \pi, \psi)]$ for all $s \in \mathbb{C}$ and some/each fixed additive character $\psi$ of $F$.

The character distributions of $\pi_n$ converge to that of $\pi$ as distributions (functionals on $C_c^\infty(G(F))$.