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 $\lambda \ge 3$. 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-Metsänkylä 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.