  The purpose of this article is proving the equality of two natural $\mathcal
L$-invariants attached to the adjoint representation of a weigth one cusp form,
each defined by purely analytic, respectively algebraic means. The proof
departs from Greenberg's definition of the algebraic $\mathcal L$-invariant as
a universal norm of a canonical $\mathbb{Z}_p$-extension of $\mathbb{Q}_p$
associated to the representation. We relate it to a certain $2\times 2$
regulator of $p$-adic logarithms of global units by means of class field
theory, which we then show to be equal to the analytic $\mathcal L$-invariant
computed by Rivero and the second author.
