### Abstract

These notes are an extended version of a talk given by the author at the conference
“Analytic Number Theory and Related Areas”, held at Research Institute for Mathematical
Sciences, Kyoto University in November 2015. We are interested in “

*L*-data”, an axiomatic framework for*L*-functions introduced by Andrew Booker in 2013 [Boo15]. 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*∈*N*then the*L*-datum is associated to a GL*(*_{n}*A*)- automorphic representation for_{F}*n*∈*N*and a number field*F*(if*F*=*Q*and there is no scaling, then*n*=*d*.). This statement was shown to be true for 0 ≤*d*< 5/3 by Booker in his pioneering paper [Boo15], and in these notes we consider an extension of his methods to 0 ≤*d*< 2. This is simultaneously a generalisation of Booker’s result and the results and techniques of Kaczorowski–Perelli in the Selberg class [KP11]. 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 [Oli16].Original language | English |
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Title of host publication | Analytic Number Theory and Related Areas |

Publisher | RIMS Kôkyûroku |

Pages | 48-59 |

Number of pages | 11 |

Volume | 2014 |

Publication status | Published - Jan 2017 |

### Publication series

Name | RIMS Kôkyûroku Bessatsu |
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Publisher | RIMS |

ISSN (Print) | 1880-2818 |

ISSN (Electronic) | 1881-6193 |

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## Cite this

Oliver, T. (2017). Notes on Low Degree L-Data. In

*Analytic Number Theory and Related Areas*(Vol. 2014, pp. 48-59). [5] (RIMS Kôkyûroku Bessatsu). RIMS Kôkyûroku. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/2014.html