  We fix an elliptic curve $E/\mathbb{F}_q(t)$ and consider the family
$\{E\otimes\chi_D\}$ of $E$ twisted by quadratic Dirichlet characters. The
one-level density of their $L$-functions is shown to follow orthogonal symmetry
for test functions with Fourier transform supported inside $(-1,1)$. As an
application, we obtain an upper bound of 3/2 on the average analytic rank. By
splitting the family according to the sign of the functional equation, we
obtain that at least $12.5\%$ of the family have rank zero, and at least
$37.5\%$ have rank one. The Katz and Sarnak philosophy predicts that those
percentages should both be $50\%$ and that the average analytic rank should be
$1/2$. We finish by computing the one-level density of $E$ twisted by Dirichlet
characters of order $\ell\neq 2$ coprime to $q$. We obtain a restriction of
$(-1/2,1/2)$ on the support with a unitary symmetry.
