Let f ∈ S_k(Γ ₀(N), ψ) be a p-ordinary newform and [special characters omitted]_f the associated Galois representation. We find the special value L_p(1, [special characters omitted]_f ⊗ [special characters omitted]). We define the analytic [special characters omitted]-invariant of a "motivic" Galois representation, and show how this special value relates to work of Greenberg and Hida on finding [special characters omitted]_p(Ad([special characters omitted]_f)). In particular, we reduce finding this value to showing an equality of p-adic L-functions similar to a well-known relation of archimedean L-functions.

Given a Hecke character ψ for an imaginary quadratic field K, let f be the theta series corresponding to ψ. We show that one has an equality L_p(s , Ad(f)) = L_p( s, [special characters omitted]) · L_p(s, ψ ⁻) corresponding to the well-known decomposition [special characters omitted](ψ) = [special characters omitted] ⊗ Ind(ψ⁻), where [special characters omitted], is the Dirichlet character corresponding to the extension K/[special characters omitted] and ψ⁻ is the "anticyclotomic part" of the character ψ. Using our computation above and theorems of Gross [Gro80] and Hida [Hid07], this leads to a formula for the [special characters omitted]-invariant of the representation Ad(f), which is exactly the value conjectured by Hida in [Hid04]. This also gives a new proof of the Ferrero-Greenberg theorem in the case of a quadratic Dirichlet character.