  We define generalised zeta functions associated to indefinite quadratic forms
of signature (g-1,1) -- and more generally, to complex symmetric matrices whose
imaginary part has signature (g-1,1) -- and we investigate their properties.
These indefinite zeta functions are defined as Mellin transforms of indefinite
theta functions in the sense of Zwegers, which are in turn generalised to the
Siegel modular setting. We prove an analytic continuation and functional
equation for indefinite zeta functions. We also show that indefinite zeta
functions in dimension 2 specialise to differences of ray class zeta functions
of real quadratic fields, whose leading Taylor coefficients at s=0 are
predicted to be logarithms of algebraic units by the Stark conjectures.
