  Given two distinct number fields $K$ and $M$, and finite order Hecke
characters $\chi$ of $K$ and $\eta$ of $M$ respectively, we say that the pairs
$(\chi, K)$ and $(\eta, M)$ are arithmetically equivalent if the associated
L-functions coincide: $$L(s, \chi, K) = L(s, \eta, M) .$$ When the characters
are trivial, this reduces to the question of fields with the same Dedekind zeta
function, investigated by Gassman in 1926, who found such fields of degree 180,
and by Perlis (1977) and others, who showed that there are no nonisomorphic
fields of degree less than $7$.
  We construct infinitely many such pairs where the fields are quadratic. This
gives dihedral automorphic forms induced from characters of different quadratic
fields. We also give a classification of such characters of order 2 for the
quadratic fields of our examples, all with odd class number.
