<p>We study characteristic functions of unbounded maximal dissipative operators in separable Hilbert spaces. Properties of maximal dissipative operators homogeneous with respect to the group of automorphisms of the open upper half-plane are established. Connections found (a) between the characteristic functions of a maximal dissipative operator and the characteristic functions of their Hermitian parts, (b) between characteristic functions of a symmetric operator corresponding distinct complex numbers from the open upper half-plane.</p>