  In this paper, we study some basic analytic properties of the boundary term
of Fesenko's two-dimensional zeta integrals. In the case of the rational number
field, we show that this term is the Laplace transform of certain infinite
series consisting of $K$-Bessel functions. It is known that the non-positivity
property of the fourth log derivative of such series is a sufficient condition
for the Riemann hypothesis of the Hasse-Weil $L$-function attached to an
elliptic curve. We show that such non-positivity is a necessary condition under
some technical assumption.
