Abstract
Let p be a prime of the form 2ℓ+1 or 4ℓ+1, where ℓ is also a prime. We prove θ(χ,i)≠0 for all primitive Dirichlet characters χ with conductor p except for the quadratic and the quartic ones. Our results generalize a theorem of Bengoechea, which asserts θ(χ,i)≠0 for non-quadratic χ with “large” prime conductor p=2ℓ+1, where ℓ is also a prime.
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