Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu,T)$ and $\mathcal{K}$ its Kronecker factor. We denote by U the restriction of T onto $\mathcal{K}^{\perp}$ the orthocomplement of $\mathcal{K}$. We give a spectral characterization in L2 of Wiener–Wintner functions in terms of the capacity of the support of the maximal spectral type of U and the almost everywhere continuity of the fractional rotated ergodic Hilbert transform. The study of the L2 case leads to new classes of dynamical systems.