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Let be a local ring, and let C be a semidualizing R-module. In this paper, we are concerned with the C-injective and G _C -injective dimensions of certain local cohomology modules of R. Firstly, the injective dimension of C and the above quantities are compared. Secondly, as an application of the above comparisons, a characterization of a dualizing module of R is given. Finally, it is shown that if R is Cohen-Macaulay of dimension d such that is C-injective, then R is Gorenstein. This is an answer to a question which was recently raised.