pn-QRPA is considered an efficient model to extract the GT strengths for the ground as well as excited states of the involved nuclei in stellar matter thanks to the huge available model space of seven major shells. The transitions from the excited states contribute effectively to the total electron capture rate and a microscopic calculation of excited state GT strength distributions is desirable. The electron capture rates play a pivotal role in the dynamics of core collapse of stars. The pn-QRPA model is used in the present work to calculate the GT strength functions and associated electron capture/$\beta^{+}$-decay rates for proton-rich nucleus $^{57}$Zn.

The Hamiltonian of the pn-QRPA is given by $$\label{GrindEQ__1_}
{\rm H}^{{\rm QRPA}} {\rm \; =\; H}^{{\rm sp}} {\rm \; +\; V}^{{\rm
pair}} {\rm \; +\; V}_{{\rm GT}}^{{\rm ph}} {\rm \; +\; V}_{{\rm
GT}}^{{\rm pp}},$$ where $H^{sp} $ is the single-particle Hamiltonian, $V^{pair} $ is the pairing force, $V_{GT}^{ph} $ is the particle-hole (ph) GT force, and $V_{GT}^{pp} $ is the particle-particle (pp) GT force. Pairing is treated in the *BCS* approximation, where an assumed constant pairing force with force strength $G$ ($G_{p}$ and $G_{n}$ for protons and neutrons, respectively) is applied, $$\begin{aligned}
 \label{GrindEQ__2_}
V^{pair} =-G\sum _{jmj'm'}(-1)^{l+j-m}  c_{jm}^{+} 