Does it exist a p-adic L function which interpolates the values of the complex one at positive integers?

Question

I known that there are classical ways to construct $p$-adic $L$ functions for Dirichlet characters through $p$-adic integrals.

We fix a character $\chi$ modulo $Np^r$ with $N$ and $p$ coprime and an integer $b$ coprime with $p$. It is possibile to create a mesure on $(\mathbb{Z}/N\mathbb{Z})^{\ast}\times \mathbb{Z}_p^{\ast}$ such that the integral of $z_p^k \xi(z)$ (for $z_p$ in $\mathbb{Z}_p^{\ast}$, $k\geq 0$ and $\xi$ a character of $(\mathbb{Z}/N\mathbb{Z})^{\ast}\times \mathbb{Z}_p^{\ast}$ of finite order) is $(1-\xi\chi_1(b)b^{k+1})L_N(-k,\xi\chi)$, where we remove the Euler factors in $N$ and $p$ from the $L$ function of the corresponding primitive character, and $\xi\chi_1$ is the restriction of the character to its $p$-part.

Is there a way to construct similar mesures which would gave $L_N(k,\xi\chi)$? If we define a mesure on $1 + p\mathbb{Z}$ by pullback of the previous mesure, we can then get a mesure which gaves $L_N(-k,\xi\chi\omega^{-k})$ for $\omega$ the Teichmüller character of $\mathbb{Z}_p$. Assuming that the character $\xi\chi\omega^{-k}$ has conductor divisible by $p$, we can easily interpolate the missing Euler factors by the $p$-adic exponential, and then by the complex functional equation it is possible to have, up to trascendental factors and a power of the conductor, $L_N(k+1,\overline{\xi\chi\omega^{-k}})$.

What can we say if the character $\xi\chi\omega^{-k}$ has conductor not divisible by $p$? And is it possible to get some algebraic part of $L_N(k,\xi\chi\omega^{-k})$ integrating $z_p^k \xi(z)$?

Because $z_p^k \xi(z)$ is the function I have to integrate, and $L_N(k,\xi\chi\omega^{-k})$ is the result I need.

I thought about composing this mesure with the inversion of $(\mathbb{Z}/N\mathbb{Z})^{\ast}\times \mathbb{Z}_p^{\ast}$, but then I should integrate $z_p^{-k}$ and I do not know what I get from it. Maybe it is well-known, but I could not find any references about these kind of integrals at negative powers.

Answer 1

The problem is that your $L_N(k,\xi\chi)$ do not lie in an extension of $\mathbb{Q}_p$.

You can interpolate the algebraic part, but using the functional equation of the original Dirichelt L-functions you can see that

$$L(k,\xi\chi)=\gamma (k)L(-k,\xi\chi)$$

where $\gamma (k)$ is a trascendental gamma factor, and obviously $L(-k,\xi\chi)\in\mathbb{Q}$

So you are really just doing the Kubota-Leopoldt construction again (Robin Chapman alredy pointed this out in the comments). The argument with p-adic measures should give the same result.