The dielectric behavior of solvent water around DNA is elicited in terms of Kirkwood-Grunwald theory from a 14 ns molecular dynamics trajectory of B-DNA developed in a medium of explicit waters and sodium counterions with particle mesh Ewald for long-range electrostatics. The computed dielectric profile near DNA, increases rather rapidly with distance and displays bulk behavior beyond 5 Angstrom. A proximity analysis of the dielectric function reveals that the relative permittivities in the first shell of DNA obey the following trend in the simulations : phosphate backbone > major groove > minor groove. Estimates of the local dielectric constants in the major groove are consistent with interpretations based on fluorescence measurements, indicating that MD models of solvent around DNA are providing a reasonably accurate account of the local solution environment of a complicated polyelectrolyte. The calculated dielectric profile is fit to a sigmoidal function, which can be used in estimating the strength of charge-charge interactions around DNA.