A mean value theorem for cubic fields

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define S(x) = ∑_n≤x r(n) and (x) = S(x) - αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of ∫₀^∞ (e^y)² e^-2yσ dy is 1/3 as expected.