There is something in common between
cluster variables in the theory of cluster algebras,
L-functions in number theory,
namely the fact that both map direct sums to products, just like determinants do.
This kind of behavior has been considered by Kapranov in "Analogies between the Langlands correspondence and topological quantum field theory".
A typical relation between cluster variables looks like $ X X' = Y + Z $, where $X$ and $X'$ are cluster variables and $Y$ and $Z$ are monomials (products of cluster variables).
Are there any known relations of this shape between L-functions ?