Algebraic differential independence concerning the Euler Γ-function and Dirichlet series

This article investigates the algebraic differential independence concerning the Euler Gamma-funciton and the function F in a certain class F which contains Dirichlet L-functions, L-functions in the extended Selberg class, or some periodic functions. We prove that the Euler and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions phi with rho(phi) < 1.