The influence of quenching temperature, heating atmosphere and annealing time on superconducting characteristics has been studied for $LuBa_2Cu_3O_{7-z}$ compound which has been recently synthesized in a nearly phase-pure form. Resistivity measurements for the as-prepared sample heated at $300^{circ}C$ in oxygen and subsequently quenched into liquid nitrogen revealed that there is no significant change in $T_c$. On the other hand, $T_c$ of the sample slightly increased when the sample was heated at $300^{circ}C$ either in air or in $N_2$ atmosphere. It was also found that $T_c$ of the sample decreased when the annealing temperature in $N_2$ atmosphere increased above $400^{circ}C$. The experimental results indicated that the as-prepared sample is under overdoped state. The enhanced superconducting transition observed in the samples after heating at $300^{circ}C$ in air or $N_2$ atmosphere was discussed in conjunction with a slight removal of oxygen and ordering of oxygen atoms in the as-prepared sample.