  An existing conjecture states that the Shannon mutual information contained
in the ground state wavefunction of conformally invariant quantum chains, on
periodic lattices, has a leading finite-size scaling behavior that, similarly
as the von Neumann entanglement entropy, depends on the value of the central
charge of the underlying conformal field theory describing the physical
properties. This conjecture applies whenever the ground state wavefunction is
expressed in some special basis (conformal basis). Its formulation comes mainly
from numerical evidences on exactly integrable quantum chains. In this paper
the above conjecture was tested for several general non-integrable quantum
chains. We introduce new families of self-dual $Z(Q)$ symmetric quantum chains
($Q=2,3,\ldots$). These quantum chains contain nearest neighbour as well
next-nearest neighbour interactions (coupling constant $p$). In the cases $Q=2$
and $Q=3$ they are extensions of the standard quantum Ising and 3-state Potts
chains, respectively. For $Q=4$ and $Q\geq 5$ they are extensions of the
Ashkin-Teller and $Z(Q)$ parafermionic quantum chains. Our studies indicate
that these models are interesting on their own. They are critical, conformally
invariant, and share the same universality class in a continuous critical line.
Moreover, our numerical analysis for $Q=2-8$ indicate that the Shannon mutual
information exhibits the conjectured behaviour irrespective if the conformally
invariant quantum chain is exactly integrable or not. For completeness we also
calculated, for these new families of quantum chains, the two existing
generalizations of the Shannon mutual information, which are based on the
R\'enyi entropy and on the R\'enyi divergence.
