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Motivated by the updated analysis of the $G(3900)$ by the BESIIIcollaboration, we perform a global analysis of the cross sections of the$e^+e^-\to D\bar{D}$, $e^+e^-\to D\bar{D}^*+c.c.$, $e^+e^-\to D^*\bar{D}^*$processes, especially focusing on the properties of the $G(3900)$. As theenergy region of interest is limited by the next opening threshold, i.e. the$D_1\bar{D}$ threshold, we focus on the energy region$[3.7,4.25]~\mathrm{GeV}$, where three charmonia $\psi(1D)$, $\psi(3S)$ and$\psi(2D)$ explicitly contribute to the cross sections. By constructing the$P$-wave contact interaction between the $(D,D^*)$ doublet and its antiparticlein the heavy quark limit, we extract the physical scattering amplitude bysolving the Lippmann-Schwinger equation. No matter whether three or twocharmonium states are included in our framework, we always find a dynamicallygenerated state corresponding to the $G(3900)$, which suggests it to be a$P$-wave dynamically generated state. We also predict several dynamicallygenerated states in the corresponding $1^{-+}$ channel. These states can befurther searched for in the electron-positron annihilation process involvingthe emission of a single photon.

The resonance parameters of the vector charmonium-like state $G(3900)$