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Abstract
For a primitive Dirichlet character χ of conductor N set θ_{χ}(τ) = ∑_{n} _{∈ℤ} n^{∈} χ(n) e^{πin2τ/N} (where ∈ = 0 for even χ, ∈ = 1 for odd χ) the associated theta series. Its value at its point of symmetry under the modular transformation τ(image found)−1/τ is related by θ_{χ}(i) = W(χ)θ(image found) (i) to the root number of the Lseries of χ and hence can be used to calculate the latter quickly if it does not vanish. Using Shimura’s reciprocity law, we calculate the Galois action on these special values of theta functions with odd N normalised by the Dedekind eta function. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the nonvanishing of these special theta values with prime N.
Original language  English 

Pages (fromto)  347360 
Number of pages  14 
Journal  Journal de Théorie des Nombres de Bordeaux 
Volume  28 
Issue number  2 
Early online date  15 Mar 2016 
DOIs  
Publication status  Published  1 Jun 2016 
Bibliographical note
© Société Arithmétique de Bordeaux, 2016. This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy. Further copying may not be permitted; contact the publisher for detailsKeywords
 Complex multiplication
 Lseries
 Shimura’s reciprocity law
 Theta functions
Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research