Special values of p-adic L-functions and Iwasawa λ-invariants of Dirichlet characters

We investigate the Iwasawa λ λ -invariant λ ₚ(χ ) λ p ( χ ) of the p -adic L -function associated with a Dirichlet character χ χ for odd primes p . Using special values of the p -adic L -function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases λ = 0 λ = 0 , λ = 1 λ = 1 , λ = 2 λ = 2 , and λ 3 λ ⩾ 3 . In particular, we study the case when the p -adic L -function vanishes at $$s=0$$ s = 0 . Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for λ ₚ(χ ) >1 λ p ( χ ) > 1 and λ ₚ(χ ) > 2 λ p ( χ ) > 2 . Furthermore, we extend the methods of Ernvall-Metsänkylä and of Dummit et al. to calculate the λ λ -invariant. This is achieved by twisting χ χ by characters ψ ψ of p -power order and using the values of the p -adic L -function at s=2-p, , 0 s = 2 - p , ⋯ , <mm