This paper focuses on a spectral property for linear time-invariant (LTI) dynamical systems represented by delay-differential equations (DDEs) entitled multiplicity-induced-dominancy (MID), which consists, roughly speaking, in the spectral abscissa of the system being defined by a multiple spectral value. More precisely, we focus on the MID property for spectral values with over-order multiplicity, i.e., a multiplicity larger than the order of the DDE. We highlight the fact that a root of over-order multiplicity is necessarily a root of a particular polynomial, called the elimination-produced-polynomial, and we address the MID property using a suitable factorization of the corresponding characteristic function involving special functions of Kummer type. Additional results and discussion are provided in the case of the $n$-th order integrator, in particular on the local optimality of a multiple root. The derived results show how the delay can be further exploited as a control parameter and are applied to some problems of stabilization of standard benchmarks with prescribed exponential decay.