  We study the exactly solvable 1D model: the dimerized $XY$ chain with uniform
and staggered transverse fields, equivalent upon fermionization to the
noninteracting dimerized Kitaev-Majorana chain with modulation. The model has
three known gapped phases with local and nonlocal (string) orders, along with
the gapless incommensurate (IC) phase in the $U(1)$ limit. The criticality is
controlled by the properties of zeros of model's partition function,
analytically continued onto the complex wave numbers. In the ground state they
become complex zeros of the spectrum of the Hamiltonian. The analysis of those
roots yields the phase diagram which contains continuous quantum phase
transitions and weaker singularities known as disorder lines (DLs) or
modulation transitions. The latter, reported for the first time in this model,
are shown to occur in two types: DLs of the first kind with continuous
appearance of the IC oscillations, and DLs of the second kind corresponding to
a jump of the wave number of oscillations. The salient property of zeros of the
spectrum is that the ground state is shown to be separable (factorized) and the
model is disentangled on a subset of the DLs. From analysis of those zeros we
also find the Majorana edge states and their wave functions.
