  Using the numerical diagonalization method, we examine the one-dimensional
t_1-t_2-J_1-J_2 model (zigzag chain t-J model) which represents an effective
model for metallic CuO double chain in the superconductor
Pr_2Ba_4Cu_7O_15-\delta. Based on the Tomonaga-Luttinger liquid theory, we
calculate the Luttinger-liquid parameter K_\rho as a function of electron
density n. It is found that superconductivity is realized in parameter region
corresponding to the experimental result. We show phase diagram of spin gap on
the t_2/|t_1|-n plane by analyzing the expectation value of twist-operator
Z_\sigma in the spin sector. The spin gap appears in the region with large
t_2/|t_1|, where the phase boundary at half-filling is consistent with that of
the known frustrated quantum spin system. The analysis also suggests that the
estimated value of the spin gap reaches 100K in the realistic parameter region
of Pr_2Ba_4Cu_7O_15-delta.
