In this work we systematically investigate the effects of the flow elasticity and inertia in polymer-induced drag reduction through (pseudo)spectral simulations of a turbulent channel flow of a dilute polymer solution. Viscoelastic effects are modeled by the finite-extensibility nonlinear elastic dumbbell model with the Peterlin approximation. The present work updates the low Weissenberg results (Weτ0⩽50) reported in earlier works by Sureshkumar et al. [Phys. Fluids 9, 743 (1997)] and Dimitropoulos et al. [J. Non-Newtonian Fluid Mech. 79, 433 (1998)] for a zero shear rate friction Reynolds number, Reτ0=125, by allowing for a lower value for the numerical diffusivity. In addition, we examine two effects on drag reduction: (A) high elasticity, by varying Weτ0 from 62.5 to 125 for a constant Reτ0=125, (B) friction Reynolds number, Reτ0=180, 395, and 590, for a constant Weτ0=50. In the high elasticity region, the mean Reynolds, Remean, continues to increase with increasing Weτ0, albeit at a smaller rate. Thus, the drag reduction achieved at the highest Weτ0 number, Weτ0=125, is about 37.5%, as compared to about 30% for Weτ0=50. On the other hand, the percent drag reduction remains virtually unchanged as Reτ0 is increased to 180, 395, and 590 for a constant Weτ0=50. Increasing the friction Reynolds number while keeping the friction Weissenberg number constant, does affect the detailed turbulent statistics. However, the boundary and the buffer layers approach an asymptote at a friction Reynolds number of 395 as it has also been observed in the Newtonian limit. The effect of variations in the computational domain, mesh resolution and the numerical diffusivity on the turbulent statistics is also reported.