Is the case of black dahlia in la noire because the game is set on 1947 Los Angeles

Yes. The whole Homicide Desk series of cases are about a series of murders that seem to be solved, but that Cole thinks are related.
In the end it is revealed that the 'real' Black Dahlia killer was behind all of these murders. Though Cole solves the case and kills the murderer, the events are covered up. So that ties in with how the real Black Dahlia murder is unsolved.