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The standard twist $F(s,\unicode[STIX]{x1D6FC})$ of $L$ -functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $\unicode[STIX]{x1D6E4}$ -factor of the $L$ -functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$ -functions. We show that the standard twist $F(s,\unicode[STIX]{x1D6FC})$ satisfies a functional equation reflecting $s$ to $1-s$ , whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of $F(s,\unicode[STIX]{x1D6FC})$ .

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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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