$p$-adic L function of an odd Dirichlet character

Question

Apologies for a naive question (especially for Iwasawa theorists): it is well-known and trivial to prove that the usual (elementary) construction of $p$-adic L functions attached to odd Dirichlet characters leads to a function which is identically zero (experts even have a highbrow explanation for this) for a "silly reason" as one expert says (essentially $\chi(-1)+1=0$). But still, isn't there any kind of $p$-adic "object" which interpolates the Euler numbers for instance, essentially $2L(\chi_{-4},-2k)$, or $3L(\chi_{-3},-2k)$, similar to Kubota--Leopoldt interpolating $\zeta(1-2k)$ after removing some $p$-Euler factor ?

Answer 1

Theorem. Let $p > 2$ be prime, and let $\chi$ be a Dirichlet character (non-trivial, and of prime-to-$p$ conductor, for simplicity). Choose $t \in \mathbb{Z} / (p-1)\mathbb{Z}$ such that $(-1)^t = \chi(-1)$. Then there exists a continuous function $L_{p, t}(\chi, -) : \mathbb{Z}_p \to \mathbb{C}_p$ such that for all integers $k \ge 0$ with $k = t \pmod {p-1}$, we have $L_{p, t}(\chi, k) = (1 - \chi(p) p^{k-1}) \cdot L(\chi, 1 - k)$.

(Edited to add: We can say much more than continuity; in fact if $\mathcal{O}$ is the ring of integers of the extension of $\mathbb{Q}_p$ generated by the values of $\chi$, we have $L_{p, t}(\chi, s) = F( (1 + p) ^ s - 1)$ for some power series $F \in \mathcal{O}[[T]]$.)

Taking a more highbrow view, one should think of $L_p(\chi)$ as an element of the Iwasawa algebra $\mathcal{O}[[\mathbb{Z}_p^\times]]$. This ring is a direct product of $(p-1)$ subrings, indexed by $\mathbb{Z} / (p-1)$, each of them isomorphic to $\mathcal{O}[[T]]$. For any $\chi$, the element $L_p(\chi)$ will project to zero in half of those components, but which half will depend on the parity of $\chi$.)

(My go-to reference for a modern account of $p$-adic Dirichlet $L$-functions is Dasgupta, Factorization of p-adic Rankin L-series, section 3.1.)

EDIT, 9.5.23. There seems to be some confusion here about the role of the Teichmüller char $\omega$. Note that there were no $\omega$'s at all in what I wrote! The interpolation formula I gave, $$L_{p, t}(\chi, k) = \star \cdot L(\chi, 1 - k) \ \text{for $k \ge 0$ st $k = t \bmod {p-1}$},$$ (where $\star$ is a local correction factor at $p$ which I want to ignore for now), is a special case of the more general statement $$L_{p, t}(\chi, k) = \star \cdot L(\chi \omega^{t - k}, 1 - k) \ \text{for all $k \ge 0$}.$$ If we set $t = 0$, then this gives the interpolation formula from Henri's remark. But the theorem is true for all $t$ with $(-1)^t = \chi(-1)$; so if $\chi$ is odd, you cannot set $t = 0$; but there are still lots of other interesting choices of $t$ you can use.