  We consider the Elliptic Ginibre Ensemble, a family of random matrix models
interpolating between the Ginibre Ensemble and the Gaussian Unitary Ensemble
and such that its empirical spectral measure converges to the uniform measure
on an ellipse. We show the convergence in law of its normalised characteristic
polynomial outside of this ellipse. Our proof contains two main steps. We first
show the tightness of the normalised characteristic polynomial as a random
holomorphic function using the link between the Elliptic Ginibre Ensemble and
Hermite polynomials. This part relies on the uniform control of the Hermite
kernel which is derived from the recent work of Akemann, Duits and Molag. In
the second step, we identify the limiting object as the exponential of a
Gaussian analytic function. The limit expression is derived from the
convergence of traces of random matrices, based on an adaptation of techniques
that were used to study fluctuations of Wigner and deterministic matrices by
Male, Mingo, P{\'e}ch{\'e} and Speicher.This work answers the interpolation
problem raised in the work of Bordenave, Chafa{\"i} and the second author of
this paper for the integrable case of the Elliptic Ginibre Ensemble and is
therefore a fist step towards the conjectured universality of this result.
