  A zero-gap state (ZGS) has been found in a bulk system of two-dimensional
organic conductor, $\alpha$-(BEDT-TTF)$_2$I$_3$ salt which consists of four
sites of donor molecules in a unit cell. In the present paper, the
characteristic of the ZGS is analyzed in detail and the electronic properties
are examined in the vicinity of the contact point where the conduction and
valence bands degenerate to form the zero-gap. The eigenvectors of the energy
band have four components of respective sites, where two of them correspond to
inequivalent sites and the other two correspond to equivalent sites. It is
shown that the former exhibits an exotic momentum dependence around the contact
point and the latter shows almost a constant dependence. The density of states
of each site close to the Fermi point is calculated to demonstrate the
temperature dependence of the local magnetic susceptibility and the local
nuclear magnetic relaxation rate. Further, the robust property of the ZGS
against the anion potential is also shown by using the second-order
perturbation.
