Pair arithmetical equivalence for quadratic fields
Given two nonisomorphic number fields K and M, and finite order Hecke characters χ of K and η of M respectively, we say that the pairs (χ, K) and (η, M) are arithmetically equivalent if the associated Lfunctions coincide: L(s,χ,K)=L(s,η,M).When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassmann in 1926, who found such fields of degree 180, and by Perlis (J Number Theory 9(3):342–360, 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.
This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect postacceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s0020902102706w
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Work Title  Pair arithmetical equivalence for quadratic fields 

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Publication Date  March 1, 2021 
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Deposited  May 27, 2022 
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