One of the main concerns about the ignition and burn of deuterium-tritium (DT) plasma is the negative effect of impurities that can get into the thermonuclear fuel during target compression on the ignition of an inertial confinement fusion (ICF) target. So, the ignition condition of the spherical DT plasma of the ICF target in the presence of high-density carbon impurities at an arbitrary concentration is found. The ignition criterion of nuclear fusion plasmas is investigated using a two-temperature model derived from a common model. The ignition criterion is described by a surface in the three-dimensional space. This surface is defined by the electron and ion temperatures, Te and Ti, and the plasma density times the hot spot dimension, ρ·R, for different concentrations of impurities. In this paper, a two-temperature laser induced shock wave is used. We found that one needs a laser intensity of 7.5 × 1021 W/cm2, a pulse duration of 1ps, and an energy of 0.44 kJ that induces a compression of κ = 4 to ignite a pure DT pre-compressed target at about 500 g/cm3. Given the constant intensity of the laser, the increase in the carbon impurity concentration increases the alpha energy deposition in the igniter zone; on the other hand, the temperature of the igniter zone decreases as a result of high losses of power densities, so that for impurity values of more than 20%, practically no hot spots are formed.