  Let $p$ be an odd prime number. Let $f$ be a normalized Hecke eigen-cuspform
that is non-ordinary at $p$. Let $K$ be an imaginary quadratic field in which
$p$ splits. We study the Artin formalism for the two-variable signed $p$-adic
$L$-functions attached to $f$ over $K$. In particular, we give evidence of a
prediction made by Castella--Ciperiani--Skinner--Sprung.
