Poles of Kloosterman Zeta Function

Question

I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

$$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)$$

and

$$Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)$$

where $S(n,m;p)$ is the Kloosterman sum. This sum converges when $s$ has a large real part and can be analytically continued to define $Z_{+/-}(s)$ on the whole complex $s$ plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether $Z_{+/-}(s)$ has poles for any integer values of $s$, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions but I have not found quite what I'm looking for. Most treatments focus on the general properties of the Kloosterman Zeta function, which are quite complicated.
My suspicion is that because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.

Answer 1

For either of them, the only pole at an integer point is at $s=1$. There is a formula for the residue in Iwaniec's book on spectral theory of automorphic forms which is too involved to write here. Note, however, that your normalization is slightly different from his ( 2$s$ instead of $s$ ). There are other poles on the line Re $s=1$ (which are related to eigenvalues of the hyperbolic Laplacian) and these are the only poles. All these come from the spectral decomposition of the zeta function.

Answer 2

I have worked through the answer using Iwaneic's book and have found that an answer which appears slightly different from that given above. In particular, I find that there is a pole for s=0,-1,-2,...

In particular I find that

1) there will be a pole at s=1 for a general Kloosterman zeta function (defined in terms of an arbitrary subgroup of SL(2,Z)). For the case of SL(2,Z) (the case of interest here) the residue happens to vanish and there is no pole.

2) the version of the Kloosterman zeta function which I defined above is slightly different from that which Iwaniec studies. Iwaniec's zeta function indeed only has a pole at s=1, but is related to the zeta function I define above by multiplication by some Gamma functions. These Gamma functions lead to simple poles at the non-positive integers.