  Goss zeta values can be found, in some cases, as evaluations of a new type of
rigid analytic function on projective curves $X$ over a finite field
$\mathbb{F}_q$, called "Pellarin $L$-series". In the case of genus $0$ and $1$,
Pellarin and Green--Papanikolas further determined functional identities for
Pellarin $L$-series, in partial analogy with the functional equation of
Dirichlet $L$-series.
  The aim of this paper is to prove that a generalization of these functional
identities holds in arbitrary genus. Our proof exploits the topological nature
of divisors on the curve $X$, as well as the introduction of an "adjoint shtuka
function". This allows us to reinterpret Pellarin $L$-series as dual versions
of the special functions studied by Angl\`es, Ngo Dac, and Tavares Ribeiro.
