This paper reports necessary stability conditions for a parabolic partial differential equation (PDE) interconnected through the boundaries to an ordinary differential equation (ODE). We intend to propose numerical certificates for the instability of such interconnections. From one side, using spectral methods, we derive a necessary analytical condition based on root locus analysis which can be tested in the parameters space. On the other side, using Lyapunov direct and converse approaches, two necessary conditions of stability are established in terms of matrix inequalities. The novelties lie both in the type of system studied and in the converse stability methods that are used. The numerical results demonstrate the performance of the different criteria set up in this paper.