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.
Iwaniec, On the order of vanishing of modular L -functions at the critical point , S\' e m
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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.