  We investigate the moment problem and Jacobi matrix associated -- by the
operator theoretic framework of the semilocal trace formula -- to each finite
set $S$ of places of $\mathbb Q$ containing the archimedean place. The measure
is given by the absolute value squared of the product over $S$ of local factors
restricted to the critical line. We treat the case $S=\{p,\infty\}$, where a
single prime $p$ is adjoined to the archimedean place. We find that all the key
ingredients such as the moments, the orthogonal polynomials and the Jacobi
matrices can be expressed as power series in terms of the parameter $q:=1/p$.
We show that the series which appear for the moments themselves are Lambert
series. The study of the $q$-series for the coefficients of the Jacobi matrix,
and for the associated orthogonal polynomials reveals an intriguing integrality
result: all those coefficients belong to the ring $\mathbb
Z[\frac{1}{\sqrt{2}}]$ obtained by adjoining $1 / \sqrt{2}$ to the ring of
integers. The main result of this paper is the conceptual explanation of this
integrality property using Catalan numbers.
