First moment of real Dirichlet L-functions with secondary terms

We will consider the problem of evaluating the first moment of quadratic Dirichlet $L$-functions. At the central point, this moment was first evaluated by Jutila, and more accurately by Soundararajan and Young, who obtained an error term of size $X^{1/2}$ (where $X$ is the size of the family). By investigating the associated double Dirichlet series and assuming the generalized Riemann hypothesis, we will show how to obtain the essentially best possible error of size $X^{1/4}$ term with a secondary term of size $X^{1/3} \log X$.