  We construct an integrable physical model of a single particle scattering
with impurities spread on a circle. The $S$-matrices of the scattering with the
impurities are such that the quantized energies of this system, coming from the
Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial
zeros of the the Riemann $\zeta(s) $ function along the axis $\Re ( s )= \half$
of the complex $s$-plane. A simple and natural generalization of the original
scattering problem leads instead to Bethe Ansatz equations whose solutions are
the non-trivial zeros of the Dirichlet $L$-functions again along the axis $\Re
(s) = \half$. The conjecture that all the non-trivial zeros of these functions
are aligned along this axis of the complex $s$-plane is known as the
Generalised Riemann Hypothesis (GRH). In the language of the scattering problem
analysed in this paper the validity of the GRH is equivalent to the
completeness of the Bethe Ansatz equations. Moreover the idea that the validity
of the GRH requires both the duality equation (i.e. the mapping $s \rightarrow
1 - s$) and the Euler product representation of the Dirichlet $L$-functions
finds additional and novel support from the physical scattering model analysed
in this paper. This is further illustrated by an explicit counterexample
provided by the solutions of the Bethe Ansatz equations which employ the
Davenport-Heilbronn function $\CD (s) $, i.e. a function whose completion
satisfies the duality equation $\chi(s) = \chi(1-s)$ but that does not have an
Euler product representation. In this case, even though there are infinitely
many solutions of the Bethe Ansatz equations along the axis $\Re (s) = \half$,
there are also infinitely many pairs of solutions away from this axis and
symmetrically placed with respect to it.
