Abscissa of convergence of a Zeta function

[Laurent Bartholdi]

Dear All,
I have a (Witten) zeta function, zeta(s) =3D sum a_n n^-s, for which I'd
like to compute the abscissa of convergence.

1) I know the first 10^7 values of a_n (only 500 of them are non-zero)
2) I have a functional equation of the form

  zeta(s) =3D sum_{all partitions P=3D(P1,...,Pk) of {1,..,5}} f_P(s)
zeta(|P1| s)...zeta(|Pk| s),

where the f_P are explicit Dirichlet polynomials; so the RHS is
polynomial in {n^-s and zeta(ns): n in N}. (I could describe more
concretely the functional equation in an example, if someone's
interested).

Using these, I can compute numerically values of the series: I iterate
the functional equation at a given value of s, using the (truncated)
series to evaluate zeta(ns) for n>1. In this way, I could estimate
that the series diverges at s=3D1.178348, while it converges at 1.17835.

QUESTION: Is there a standard method to determine (algebraically) the
abscissa of convergence? Should I expect it to be an algebraic (or
even rational) number?

--=20
Laurent Bartholdi \ laurent.bartholdi<at>gmail<dot>com
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