For the Dirichlet series F(s)=∑n=1∞fnexp{sλn}, which is the Hadamard composition of the genus m of similar Dirichlet series Fj(s) with the same exponents, the growth with respect to the function G(s) given as the Dirichlet series is studied in terms of the Φ-type (the upper limit of MG−1(MF(σ))/Φ(σ) as σ↑A) and convergence Φ-class defined by the condition ∫σ0AΦ′(σ)MG−1(MF(σ))Φ2(σ)dσ<+∞, where MF(σ) is the maximum modulus of the function F at an imaginary line and A is the abscissa of the absolute convergence.