This talk will give an analytic construction of a $p$-adic $L$-function
associated to the "evil twin" of a classical ordinary eigenform.
A
dichotomy arises out of this construction: the critical
values of the $p$-adic $L$-function either (1) all vanish, or (2) agree
with the complex $L$-values up to standard multipliers. I will
explain
how this is related to semi-simplicity of the Hecke algebra acting
on
the
overconvergent cohomolgy. Moreover, I will describe joint work
with
Robert
Pollack, in which we establish a sufficient condition for the latter
conclusion in terms of modular forms modulo $p$. I will
also present
some intriquing numerical data on the disribution of the zeroes of
these
$p$-adic $L$-functions that arose out of our work.