## Abstract

These notes are an extended version of a talk given by the author at the confer‐

ence Analytic Number Theory and Related Areas held at Research Institute for Math‐

ematical Sciences, Kyoto University in November 2015. We are interested in L‐data, an

axiomatic framework for L\sim‐functions introduced by Andrew Booker in 2013 [3]. Associated

to each L‐datum, one has a real number invariant known as the degree. Conjecturally the

degree d is an integer, and if d\in \mathrm{N} then the L‐datum is that of a \mathrm{G}\mathrm{L}_{n}(\mathrm{A}_{F}) ‐automorphic

representation for n\in \mathrm{N} and a number field F (if F=\mathbb{Q} ,

then n=d This statement was

shown to be true for 0\displaystyle \leq d<\frac{5}{3} by Booker in his pioneering paper [3], and in these notes we

consider an extension of his methods to 0\leq d<2. This is simultaneously a generalisation

of Booker’s result and the results and techniques of Kaczorowski‐Pereli in the Selberg class

[10]. Furthermore, we consider applications to zeros of automorphic L-‐functions. In these

notes we review Booker’s results and announce new ones to appear elsewhere shortly

ence Analytic Number Theory and Related Areas held at Research Institute for Math‐

ematical Sciences, Kyoto University in November 2015. We are interested in L‐data, an

axiomatic framework for L\sim‐functions introduced by Andrew Booker in 2013 [3]. Associated

to each L‐datum, one has a real number invariant known as the degree. Conjecturally the

degree d is an integer, and if d\in \mathrm{N} then the L‐datum is that of a \mathrm{G}\mathrm{L}_{n}(\mathrm{A}_{F}) ‐automorphic

representation for n\in \mathrm{N} and a number field F (if F=\mathbb{Q} ,

then n=d This statement was

shown to be true for 0\displaystyle \leq d<\frac{5}{3} by Booker in his pioneering paper [3], and in these notes we

consider an extension of his methods to 0\leq d<2. This is simultaneously a generalisation

of Booker’s result and the results and techniques of Kaczorowski‐Pereli in the Selberg class

[10]. Furthermore, we consider applications to zeros of automorphic L-‐functions. In these

notes we review Booker’s results and announce new ones to appear elsewhere shortly

Original language | English |
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Title of host publication | RIMS Kokuroku |

Subtitle of host publication | Analytic Number Theory and Related Areas |

Editors | Yuichi Kamiya |

Publisher | Kyoto University |

Pages | 48-58 |

Publication status | Published - 2017 |