The single-particle dispersion, Lagrangian structure functions and Lagrangian energy spectra characteristic of two-dimensional incompressible turbulent flows are investigated theoretically and numerically. The domain of validity of the classical asymptotic estimates is extended; it is shown in particular that the asymptotic behavior of the single-particle dispersion at small times remains valid throughout the whole self-similar range when the Lagrangian energy spectrum is steeper than ω−1. Straightforward estimates of the Lagrangian integral time scale TL and the diffusion coefficient at large times K, based on energy and enstrophy, are proposed; to some extent, they remain valid locally, which allows an analysis of the spatial variability of TL and K within a single turbulent field. Finally, the detrimental effect of artificial numerical diffusion on the numerical simulation of Lagrangian statistics is highlighted and discussed.