  The connection zeta function of a finite abstract simplicial complex G is
defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of
the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else.
(I) As a consequence of the spectral formula chi(G)=sum_x (-1)^dim(x) =
p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the
number of negative eigenvalues of L, both the Euler characteristic
chi(G)=zeta(0)-2 i zeta'(0)/pi as well as determinant det(L)=e^zeta'(0)/pi can
be written in terms of zeta. (II) As a consequence of the generalized
Cauchy-Binet formula for the coefficients of the characteristic polynomials of
a product of matrices we show that for every one-dimensional simplicial complex
G, the functional equation zeta(s)=zeta(-s) holds, where zeta(s) is the Zeta
function of the positive definite squared connection operator L^2 of G.
Equivalently, the spectrum sigma of the integer matrix L^2 for a 1-dimensional
complex always satisfies the symmetry sigma = 1/sigma and the characteristic
polynomial of L^2 is palindromic. The functional equation extends to products
of one-dimensional complexes. (III) Explicit expressions for the spectrum of
circular connection Laplacian lead to an explicit entire zeta function in the
Barycentric limit. The situation is simpler than in the Hodge Laplacian H=D^2
case where no functional equation was available. In the connection Laplacian
case, the limiting zeta function is a generalized hypergeometric function which
for an integer s is given by an elliptic integral over the real elliptic curve
w^2=(1+z)(1-z)(z^2-4z-1), which has the analytic involutive symmetry (z,w) to
(1/z,w/z^2).
