Vanishing and non-vanishing theta values

For a primitive Dirichlet character χ of conductor N set (Formula presented.) (where ϵ= 0 for χ even, ϵ= 1 for χ odd), the value of the associated theta series at its point of symmetry under the modular transformation τ→ - 1 / τ. These numbers are related by Θ (χ) = W(χ) Θ (χ¯) to the root number of the L-series of χ and hence can be used to calculate the latter quickly if they do not vanish. We describe experiments showing that Θ (χ) ≠ 0 for all χ with N≤ 52 , 100 (roughly 500 million primitive characters) except for precisely two characters (up to χ→ χ¯), of conductors 300 and 600. The proof that Θ (χ) vanishes in these two cases uses properties of Ramanujan’s modular function of level 5. We also characterize all χ for which W(χ) is a root of unity and describe some experimental results concerning the algebraic numbers Θ (χ) / η(i) 1 + 2 ϵ when N is prime. © 2013, Fondation Carl-Herz and Springer International Publishing Switzerland.