For prime conductors over F_q[x], the paper obtains the g^2 term in the second moment of quadratic Dirichlet L-functions and the leading term in the mean derivative of elliptic-curve twists, implying a rank-one twist exists.
Moments of quadratic twists of elliptic curve L-functions over function fields
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abstract
We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.
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Moments of Dirichlet $L$-functions with prime conductors over function fields
For prime conductors over F_q[x], the paper obtains the g^2 term in the second moment of quadratic Dirichlet L-functions and the leading term in the mean derivative of elliptic-curve twists, implying a rank-one twist exists.