In the talk we present some recent results on the connection between turnpike behaviors and strict dissipativity properties for finite dimensional linear quadratic (LQ) optimal control problems. The focus is on providing characterizations of strict dissipativity and the newly introduced property of strict predissipativity in terms of the system matrices related to the LQ problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. In particular, the techniques allow to encompass the presence of state and input constraints. Finally, several examples illustrate the different turnpike behaviors of the system in the presence or absence of constraints.