<p>We consider a one-dimensional problem of propagation of a narrowband piecewise holomorphic signal in a homogeneous dispersive medium. Within the method of moments, estimates are obtained for the energy of the precursor, its average duration, and the average time of the lag of the main part of the precursor from its leading edge, which propagates at the vacuum speed of light. It is shown that in the case of a sufficient path length, when only the precursor remains of the signal, the precursor duration and time of the lag from the leading edge no longer change with increasing path length, depend only on the rank (order of discontinuity) of the original signal, and are approximately equal to the time of information storage in the medium. In the cases of a plasma and a Lorentz spectral line, this time is equal to the inverse collision frequency or the coherence time of the spectral line, respectively.</p>