We drive a 1D-Quantum transverse field Ising Model (TFIM) out of equilibrium by performing a quantum quench inside both the paramagnetic phase and the ferromagnetic one, toward the critical point and through it. A perturbation is added to the system to deduce the asymptotic properties of the transverse magnetization stationary state by the analysis of the spectral function. We find a relevant result in the case of a quench across the critical point. In this case the spectral function changes sign in correspondence of the frequencies at which TFIM exhibits a dynamical quantum phase transition (DQPT). The importance of this result lies in the link between the zeros of the spectral function and the DQPTs that gives an important hint for the experimental evaluation of the DQPTs.