We study the 1D nonlinear Schrödinger equation with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global dispersive solutions, which are constant with respect to space. The non-existence of global solutions has also been studied only by focusing on the behavior of the Fourier 0 mode of solutions. However, the earlier works are not sufficient to obtain the precise criteria for the global existence for the Cauchy problem. In this paper, the exact criteria for the global existence of L2 solutions are shown by studying the interaction between the Fourier 0 mode and oscillation of solutions. Namely, L2 solutions are shown a priori not to exist globally if they are different from the trivial ones.