We fix an elliptic curve E/Fq(t) and consider the family {E ⊗ χ 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 = 2 coprime to q. We obtain a restriction of (−1/2, 1/2) on the support with a unitary symmetry.

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We investigate in this paper the vanishing at s = 1 of the twisted L-functions of elliptic curves E defined over the rational function field F q (t) (where F q is a finite field of q elements and characteristic ≥ 5) for twists by Dirichlet characters of prime order ℓ ≥ 3, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of [Li18, DL21] who proved vanishing at s = 1/2 for infinitely many Dirichlet L-functions over F q (t) based on the existence of one, and we can prove that if there is one χ 0 such that L(E, χ 0 , 1) = 0, then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over F q (t) behave differently than the general ones.

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