  For a scalar matrix $\mathbf a=(a_{m, n})_{m, n=1}^{\infty},$ the Dirichlet
series kernel $\kappa_{\mathbf a}$ is the double Dirichlet series
$\kappa_{\mathbf a}(s, u) = \sum_{m, n =1}^{\infty} a_{m, n}m^{-s}
n^{-\bar{u}}$ in the variables $s$ and $\bar{u},$ which is regularly convergent
on some right half-plane $\mathbb H_\rho.$ The analytic symbols $A_{n, \mathbf
a} = \sum_{m=1}^{\infty}a_{m, n}m^{-s},$ $n \geq 1$ play a central role in the
study of the reproducing kernel Hilbert space $\mathscr H_{\mathbf a}$
associated with the positive semi-definite kernel $\kappa_{\mathbf a}.$ In
particular, they form a total subset of $\mathscr H_{\mathbf a}$ and provide
the formula $\sum_{n=1}^{\infty}\langle f, A_{n, \mathbf a} \rangle n^{-s},$ $s
\in \mathbb H_\rho,$ for $f \in \mathscr H_{\mathbf a}.$ We combine the basic
theory of Dirichlet series kernels with the Gelfond-Schneider theorem
(Hilbert's seventh problem) to show that any quasi-invariant Dirichlet series
kernel $\kappa_{\mathbf a}(s, u)$ factors as $f(s)\bar{f(u)}$ for some
Dirichlet series $f$ on $\mathbb H_\rho.$ In particular, there is no
quasi-invariant Dirichlet series kernel $\kappa_{\mathbf a}$ if the dimension
of $\mathscr H_{\mathbf a}$ is bigger than one. This is in strict contrast with
the case of the unit disc, where non-factorable quasi-invariant kernels exist
in abundance. We further discuss the Dirichlet series kernels $\kappa_{\mathbf
a}$ invariant under the group $\mathscr T$ of translation automorphisms of
$\mathbb H_\rho$ and construct a family of densely defined $\mathscr
T$-homogeneous operators in $\mathscr H_{\mathbf a},$ whose adjoints are
defined only at the zero vector.
