  We consider an elliptic self-adjoint first order differential operator acting
on pairs (2-columns) of complex-valued half-densities over a connected compact
3-dimensional manifold without boundary. The principal symbol of our operator
is assumed to be trace-free. We study the spectral function which is the sum of
squares of Euclidean norms of eigenfunctions evaluated at a given point of the
manifold, with summation carried out over all eigenvalues between zero and a
positive lambda. We derive an explicit two-term asymptotic formula for the
spectral function as lambda tends to plus infinity, expressing the second
asymptotic coefficient via the trace of the subprincipal symbol and the
geometric objects encoded within the principal symbol - metric, torsion of the
teleparallel connection and topological charge. We then address the question:
is our operator a massless Dirac operator on half-densities? We prove that it
is a massless Dirac operator on half-densities if and only if the following two
conditions are satisfied at every point of the manifold: a) the subprincipal
symbol is proportional to the identity matrix and b) the second asymptotic
coefficient of the spectral function is zero.
