  The low-energy properties of the elastic $s-$wave scattering for the
$n-^{19}$C are studied near the critical condition for the occurrence of an
excited Efimov state in $n-n-^{18}$C. It is established to which extent the
universal scaling laws, strictly valid in the zero-range limit, survive when
finite range potentials are considered. By fixing the two-neutrons separation
energy in $^{20}$C with available experimental data, it is studied the scaling
of the real ($\delta_0^R$) and imaginary parts of the $s-$wave phase-shift with
the variation of the $n-^{18}$C binding energy. We obtain some universal
characteristics given by the pole-position of $k\cot(\delta_0^R)$ and
effective-range parameters. By increasing the $n-^{18}$C binding energy, it was
verified that the excited state of $^{20}$C goes to a virtual state, resembling
the neutron-deuteron behavior in the triton. It is confirmed that the
analytical structure of the unitary cut is not affected by the range of the
potential or mass asymmetry of the three-body system.
