  We study the Shannon and R\'enyi mutual information (MI) in the ground state
(GS) of different critical quantum spin chains. Despite the apparent basis
dependence of these quantities we show the existence of some particular basis
(we will call them conformal basis) whose finite-size scaling function is
related to the central charge
  $c$ of the underlying conformal field theory of the model.
  In particular, we verified that for large index $n$, the MI of a subsystem of
size $\ell$ in a periodic chain with $L$ sites behaves as
$\frac{c}{4}\frac{n}{n-1}\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi
\ell}{L})\Big{)}$, when the ground-state wavefunction is expressed in these
special conformal basis. This is in agreement with recent predictions. For
generic local basis we will show that, although in some cases
$b_n\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$ is a good fit to
our numerical data, in general there is no direct relation between $b_n$ and
the central charge of the system. We will support our findings with detailed
numerical calculations for the transverse field Ising model, $Q=3,4$ quantum
Potts chain, quantum Ashkin-Teller chain and the XXZ quantum chain. We will
also present some additional results of the Shannon mutual information ($n=1$),
for the parafermionic $Z_Q$ quantum chains with $Q=5,6,7$ and $8$.
