New scenario of irreversibility for linear systems has been found and discussed. This scenario is based on the interpretation of the geometrical/physical meaning of the temporal fractional integral with complex and real fractional exponents. It has been shown that imaginary part of the fractional integral related to discrete-scale invariance (DSI) phenomenon and observed only for true regular (discrete) fractals. Numerical experiments show that the imaginary part of the complex fractional exponent can be well approximated by simple and finite combination of the leading sine/cosine log-periodical functions with period lnξ ( ξ is a scaling parameter). In the most cases analyzed the leading Fourier components give a pair of complex conjugated exponents defining the imaginary part of the complex fractional integral. For random fractals, where invariant scaling properties are realized only in the statistical sense the imaginary part of the complex exponent is averaged and the result is expressed in the form of the conventional Riemann-Liouville integral. The conditions for realization of reind and recaps elements with complex power-law exponents have been found. The fractal structures leading to pure log-periodic oscillations related to fractional integration with complex exponent are analyzed. Description of relaxation processes by kinetic equations containing complex fractional exponent and their possible recognition in the dielectric spectroscopy is discussed.