We present measurements of thermally-induced oblique-roll traveling-wave (TW) fluctuations below the supercritical primary bifurcation to electroconvection (EC) in the nematic liquid crystal 4-ethyl-2-fluoro-4'-[2-(trans-4-pentylcyclohexyl)ethyl]-biphenyl (I52). First we analyze time sequences of one-dimensional shadowgraph images taken parallel to the director to obtain the TW frequency omega and the fluctuation lifetime tau. Within our resolution we find that omega is independent of epsilon [triple bond] V/V(c)-1 (V is the applied voltage amplitude and V(c) its value at the onset of convection). Contrary to linear theory, the relaxation rate 1/tau remains finite at the bifurcation. Next we present the analysis of temporally uncorrelated two-dimensional shadowgraph images of the fluctuations for several values of the electrical conductivity sigma. We fitted an anisotropic two-dimensional Lorentzian function, corresponding to oblique-roll EC, to the time-averaged structure factors S(k) derived from the images. This yielded information about the components of the mean wave vector k(0) and about the correlation length xi as a function of sigma and epsilon. The angle of obliqueness theta of the roll patterns was independent of sigma but decreased anomalously as epsilon approached zero. The modulus k(0) of k(0) depended on sigma. It also showed an anomalous reduction close to onset. The anomalous epsilon dependence of k(0) and theta disagrees with linear theory, which predicts a smooth, essentially linear dependence on epsilon, and presumably is caused by nonlinear interactions between the fluctuations.