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

We investigate the Iwasawa $λ$-invariant $λ_p(χ)$ of the $p$-adic $L$-function associated with a Dirichlet character $χ$ 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 $