A note on the invariants of the $L$-functions

We explain the exact meaning of a statement we made in a previous paper on invariants, namely that a complex-valued function of the data of the functional equation of an $L$-function is an invariant if and only if it is stable under the multiplication and factorial formulae. To this end, we show that every invariant has a so-called rational extension, having some desired invariance properties. The existence of such an extension, which enables to express formally the above heuristic concept, is not apparent and its construction is the main novelty of the paper.