The ψ(d,D)-weighted first moment of L′(1/2, f×χ_d) over d∈D equals c_{q,D}G(q,D)J̃(1)L(1,χ_D)X log X + O(X(log X)^{1/2}(log log X)^3), with explicit positive constants.
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First Moment of derivatives of $L$-functions in a nonlinear family
The ψ(d,D)-weighted first moment of L′(1/2, f×χ_d) over d∈D equals c_{q,D}G(q,D)J̃(1)L(1,χ_D)X log X + O(X(log X)^{1/2}(log log X)^3), with explicit positive constants.