  In this expository note we show the inception and development of the
Heilbronn characters and their application to the holomorphy of quotients of
Artin L-functions. Further we use arithmetic Heilbronn characters introduced by
Wong, to deal with holomorphy of quotients of certain L-functions, e.g,,
L-functions associated to CM elliptic curves. Furthermore we use the
supercharacter theory introduced by Diaconis and Isaacs to study Artin
L-functions associated to such characters. We conclude the note surveying about
various other unconditional approaches taken based on character theory of
finite groups.
