Warning: Some naive, speculative questions from a total non expert.

Let $\rho$ be a p-adic representation of the Galois group $Gal(\overline K/K)$ for a number field $K$. We can consider the Artin L-function $L(s,\rho)$ associated to the representation. Let me define $L\_p(s,\rho)$ to be $L(s,\rho)$ with the Euler factor at (primes above) $p$ removed. 

Now, in certain special cases we have congruences of the form:
$$L\_p(1-n,\rho\_1) \equiv L\_p(1-n,\rho\_2)\pmod{\pi}$$
for $n \geq 1$ if $\rho\_1 \equiv \rho\_2 \pmod{\pi}$. This congruence might not even make sense in the general case (what is the reduction of a transcendental number?) but at least for, say, Dirichlet characters, these are just a form of the Kummer congruences (if we place some restrictions on what $\rho\_1,\rho\_2$ can be).

We can think of this congruence as indicating that something like the following must be true:

"It is possible to define a mod p Artin function $\overline L(s,\overline\rho)$ that depends only on the reduction $\overline\rho$ of the reduction."

Of course this is very ambitious and very ill defined in general. Nevertheless, my questions are as follows:

1) What generalizations of the Kummer congruences are known? This is probably the same question as asking "what definitions of p-adic L functions are known?". I know that it is possible to prove such results for totally real number fields but do we know of any other cases? I am interested in general L-functions here, not just Artin L-functions.

2) Have there been any attempts to actually define a mod p Artin function of the kind indicated above? Even in very simple cases (say for the Riemann zeta function), I have not come across any indication of such an attempt (which is not saying much).

A naive idea might be to just take a power series representation of a p-adic L function and reduce each term formally. For instance, let $\chi$ be a Dirichlet character, $L\_p(s,\chi) = a\_{-1}/(s-1) + a\_0 + a\_1(s-1) + \dots$ be the corresponding p-adic L-series with $a\_n \in \mathbb Q\_p(\chi)$. However, for this to even have a hope of working, the coefficients $a\_n$ of the Taylor series should be p-adic integers instead of just arbitrary p-adic numbers. I suspect this is false...