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

We study the Iwasawa $\lambda$-invariant of Dirichlet characters $\chi$ of arbitrary order for odd primes $p$. From special values of the $p$-adic $L$-function and its derivative we derive several novel and easily computable criteria to distinguish between the cases $\lambda = 0$, $\lambda = 1$, $\lambda = 2$ and $\lambda \geq 3$. In particular, we look at the case when the $p$-adic $L$-function vanishes at $s=0$. Using formulas of Ferrero-Greenberg and Gross-Koblitz, we give conditions for $\lambda_p(\chi) >1$ and $\lambda_p(\chi) > 2$. Furthermore, we extend methods of Ernvall-Mets\"ankyl\"a and Dummit et al. to calculate the $\lambda$-invariant by twisting $\chi$ with characters $\psi$ of the second kind and using the values of the $p$-adic $L$-function at $s=2-p, \dots, 0$. In addition, we leverage the value at $s=1$ to compute $\lambda_p(\chi)$. The formulas are also used to obtain numerical data on the distribution of $\lambda$-invariants, where either the prime $p$ or the Dirichlet character $\chi$ is fixed.