# lfunc_search downloaded from the LMFDB on 02 May 2026. # Search link: https://www.lmfdb.org/L/1/693/693.464 # Query "{'degree': 1, 'conductor': 693}" returned 180 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "1-693-693.104-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.104" [[0, 0.0]] [] 0 true true false false -0.12950509215261544 0 0.173456981876 ["Character/Dirichlet/693/104"] "1-693-693.128-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.128" [[0, 0.0]] [] 0 true true false false 0.137908758383297 0 1.06771303236 ["Character/Dirichlet/693/128"] "1-693-693.13-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.13" [[0, 0.0]] [] 0 true true false false -0.37572135505044374 0 0.133563122019 ["Character/Dirichlet/693/13"] "1-693-693.130-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.130" [[0, 0.0]] [] 0 true true false false -0.4916924954854688 0 1.36703674585 ["Character/Dirichlet/693/130"] "1-693-693.139-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.139" [[0, 0.0]] [] 0 true true false false 0.016711170745212833 0 1.35365301087 ["Character/Dirichlet/693/139"] "1-693-693.146-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.146" [[0, 0.0]] [] 0 true true false false -0.23156151487994953 0 0.764522380324 ["Character/Dirichlet/693/146"] "1-693-693.149-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.149" [[0, 0.0]] [] 0 true true false false 0.108030053799639 0 0.71899753871 ["Character/Dirichlet/693/149"] "1-693-693.16-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.16" [[0, 0.0]] [] 0 true true false false 0.4916924954854688 0 0.43460495469 ["Character/Dirichlet/693/16"] "1-693-693.160-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.160" [[0, 0.0]] [] 0 true true false false 0.37572135505044374 0 1.3105036284 ["Character/Dirichlet/693/160"] "1-693-693.178-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.178" [[0, 0.0]] [] 0 true true false false 0.25556270911461354 0 1.32535599226 ["Character/Dirichlet/693/178"] "1-693-693.185-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.185" [[0, 0.0]] [] 0 true true false false 0.3719002674287838 0 0.0741321690502 ["Character/Dirichlet/693/185"] "1-693-693.2-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.2" [[0, 0.0]] [] 0 true true false false -0.24547623258764048 0 1.00903013169 ["Character/Dirichlet/693/2"] "1-693-693.20-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.20" [[0, 0.0]] [] 0 true true false false 0.12950509215261544 0 1.37196911932 ["Character/Dirichlet/693/20"] "1-693-693.200-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.200" [[0, 0.0]] [] 0 true true false false -0.108030053799639 0 0.708045720015 ["Character/Dirichlet/693/200"] "1-693-693.214-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.214" [[0, 0.0]] [] 0 true true false false -0.3618137909018107 0 1.72430229742 ["Character/Dirichlet/693/214"] "1-693-693.236-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.236" [[0, 0.0]] [] 0 true true false false -0.3719002674287838 0 1.5438508735 ["Character/Dirichlet/693/236"] "1-693-693.241-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.241" [[0, 0.0]] [] 0 true true false false 0.40906184194283507 0 1.62262535953 ["Character/Dirichlet/693/241"] "1-693-693.247-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.247" [[0, 0.0]] [] 0 true true false false 0.047248051041024294 0 1.05251023744 ["Character/Dirichlet/693/247"] "1-693-693.25-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.25" [[0, 0.0]] [] 0 true true false false -0.40831465807358924 0 0.752123917321 ["Character/Dirichlet/693/25"] "1-693-693.250-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.250" [[0, 0.0]] [] 0 true true false false -0.2035507904658256 0 0.806900301 ["Character/Dirichlet/693/250"] "1-693-693.256-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.256" [[0, 0.0]] [] 0 true true false false 0.14724089748196634 0 1.00831624486 ["Character/Dirichlet/693/256"] "1-693-693.257-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.257" [[0, 0.0]] [] 0 true true false false 0.04472189515933673 0 1.26984010189 ["Character/Dirichlet/693/257"] "1-693-693.263-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.263" [[0, 0.0]] [] 0 true true false false 0.0454690790285825 0 1.40699931051 ["Character/Dirichlet/693/263"] "1-693-693.268-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.268" [[0, 0.0]] [] 0 true true false false -0.14724089748196634 0 0.799588528592 ["Character/Dirichlet/693/268"] "1-693-693.283-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.283" [[0, 0.0]] [] 0 true true false false -0.2743159954690445 0 0.816863699996 ["Character/Dirichlet/693/283"] "1-693-693.290-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.290" [[0, 0.0]] [] 0 true true false false 0.36284557904500675 0 1.50670373391 ["Character/Dirichlet/693/290"] "1-693-693.292-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.292" [[0, 0.0]] [] 0 true true false false -0.25556270911461354 0 0.798956990513 ["Character/Dirichlet/693/292"] "1-693-693.304-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.304" [[0, 0.0]] [] 0 true true false false -0.3520047650897299 0 0.616617988788 ["Character/Dirichlet/693/304"] "1-693-693.311-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.311" [[0, 0.0]] [] 0 true true false false 0.010833660396218837 0 1.41974106453 ["Character/Dirichlet/693/311"] "1-693-693.32-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.32" [[0, 0.0]] [] 0 true true false false 0.10102463458413806 0 1.06317259218 ["Character/Dirichlet/693/32"] "1-693-693.326-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.326" [[0, 0.0]] [] 0 true true false false 0.28440247199601754 0 1.30170944968 ["Character/Dirichlet/693/326"] "1-693-693.335-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.335" [[0, 0.0]] [] 0 true true false false -0.18506064770817102 0 0.766874235274 ["Character/Dirichlet/693/335"] "1-693-693.338-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.338" [[0, 0.0]] [] 0 true true false false -0.28440247199601754 0 0.883250511624 ["Character/Dirichlet/693/338"] "1-693-693.347-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.347" [[0, 0.0]] [] 0 true true false false 0.24547623258764048 0 1.77935142976 ["Character/Dirichlet/693/347"] "1-693-693.349-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.349" [[0, 0.0]] [] 0 true true false false -0.016711170745212833 0 1.59362849958 ["Character/Dirichlet/693/349"] "1-693-693.356-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.356" [[0, 0.0]] [] 0 true true false false 0.23156151487994953 0 1.52516749089 ["Character/Dirichlet/693/356"] "1-693-693.376-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.376" [[0, 0.0]] [] 0 true true false false -0.11811653032661207 0 0.767107557787 ["Character/Dirichlet/693/376"] "1-693-693.38-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.38" [[0, 0.0]] [] 0 true true false false 0.0017789720124417928 0 1.37333234651 ["Character/Dirichlet/693/38"] "1-693-693.380-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.380" [[0, 0.0]] [] 0 true true false false 0.05247449824408344 0 1.42394261319 ["Character/Dirichlet/693/380"] "1-693-693.383-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.383" [[0, 0.0]] [] 0 true true false false 0.0017789720124417928 0 0.905426896879 ["Character/Dirichlet/693/383"] "1-693-693.391-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.391" [[0, 0.0]] [] 0 true true false false -0.06872308939400074 0 0.559957684457 ["Character/Dirichlet/693/391"] "1-693-693.394-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.394" [[0, 0.0]] [] 0 true true false false -0.19374176465374485 0 0.864080134094 ["Character/Dirichlet/693/394"] "1-693-693.4-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.4" [[0, 0.0]] [] 0 true true false false -0.44519162831369025 0 0.732360150525 ["Character/Dirichlet/693/4"] "1-693-693.40-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.40" [[0, 0.0]] [] 0 true true false false -0.17012844897539994 0 0.888197513253 ["Character/Dirichlet/693/40"] "1-693-693.409-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.409" [[0, 0.0]] [] 0 true true false false -0.18888173532983094 0 1.14042353452 ["Character/Dirichlet/693/409"] "1-693-693.416-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.416" [[0, 0.0]] [] 0 true true false false 0.31634471187322827 0 1.79141092913 ["Character/Dirichlet/693/416"] "1-693-693.439-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.439" [[0, 0.0]] [] 0 true true false false 0.03538260250160944 0 1.17025317684 ["Character/Dirichlet/693/439"] "1-693-693.445-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.445" [[0, 0.0]] [] 0 true true false false 0.19374176465374485 0 1.37103172289 ["Character/Dirichlet/693/445"] "1-693-693.464-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.464" [[0, 0.0]] [] 0 true true false false -0.198968211856804 0 0.594216539265 ["Character/Dirichlet/693/464"] "1-693-693.466-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.466" [[0, 0.0]] [] 0 true true false false 0.0007471838692457777 0 0.408129479864 ["Character/Dirichlet/693/466"] "1-693-693.47-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.47" [[0, 0.0]] [] 0 true true false false -0.4184011346005623 0 0.212583223145 ["Character/Dirichlet/693/47"] "1-693-693.472-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.472" [[0, 0.0]] [] 0 true true false false -0.03538260250160944 0 0.42649926912 ["Character/Dirichlet/693/472"] "1-693-693.475-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.475" [[0, 0.0]] [] 0 true true false false 0.06872308939400074 0 1.3212081524 ["Character/Dirichlet/693/475"] "1-693-693.481-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.481" [[0, 0.0]] [] 0 true true false false 0.3520047650897299 0 1.32503707077 ["Character/Dirichlet/693/481"] "1-693-693.482-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.482" [[0, 0.0]] [] 0 true true false false 0.18506064770817102 0 1.19639920186 ["Character/Dirichlet/693/482"] "1-693-693.488-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.488" [[0, 0.0]] [] 0 true true false false -0.010833660396218837 0 1.28182460429 ["Character/Dirichlet/693/488"] "1-693-693.499-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.499" [[0, 0.0]] [] 0 true true false false 0.40831465807358924 0 1.50652789291 ["Character/Dirichlet/693/499"] "1-693-693.5-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.5" [[0, 0.0]] [] 0 true true false false -0.31634471187322827 0 0.911115252506 ["Character/Dirichlet/693/5"] "1-693-693.500-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.500" [[0, 0.0]] [] 0 true true false false 0.05733452756799736 0 0.914653610667 ["Character/Dirichlet/693/500"] "1-693-693.502-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.502" [[0, 0.0]] [] 0 true true false false 0.2743159954690445 0 1.76345300686 ["Character/Dirichlet/693/502"] "1-693-693.509-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.509" [[0, 0.0]] [] 0 true true false false -0.36284557904500675 0 0.0148832940129 ["Character/Dirichlet/693/509"] "1-693-693.52-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.52" [[0, 0.0]] [] 0 true true false false 0.17012844897539994 0 0.766272914362 ["Character/Dirichlet/693/52"] "1-693-693.520-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.520" [[0, 0.0]] [] 0 true true false false 0.44519162831369025 0 1.86484045709 ["Character/Dirichlet/693/520"] "1-693-693.527-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.527" [[0, 0.0]] [] 0 true true false false -0.0454690790285825 0 1.38092828059 ["Character/Dirichlet/693/527"] "1-693-693.535-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.535" [[0, 0.0]] [] 0 true true false false 0.2035507904658256 0 0.930494412492 ["Character/Dirichlet/693/535"] "1-693-693.536-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.536" [[0, 0.0]] [] 0 true true false false -0.137908758383297 0 0.740049679669 ["Character/Dirichlet/693/536"] "1-693-693.538-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.538" [[0, 0.0]] [] 0 true true false false 0.2222222222222222 0 1.2050653284 ["Character/Dirichlet/693/538"] "1-693-693.542-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.542" [[0, 0.0]] [] 0 true true false false -0.04472189515933673 0 0.693760830362 ["Character/Dirichlet/693/542"] "1-693-693.556-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.556" [[0, 0.0]] [] 0 true true false false -0.4374390252289435 0 0.678997568238 ["Character/Dirichlet/693/556"] "1-693-693.569-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.569" [[0, 0.0]] [] 0 true true false false -0.3399580275515731 0 0.0686527380661 ["Character/Dirichlet/693/569"] "1-693-693.578-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.578" [[0, 0.0]] [] 0 true true false false 0.198968211856804 0 1.20332729974 ["Character/Dirichlet/693/578"] "1-693-693.58-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.58" [[0, 0.0]] [] 0 true true false false 0.0007471838692457777 0 1.04966757226 ["Character/Dirichlet/693/58"] "1-693-693.580-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.580" [[0, 0.0]] [] 0 true true false false -0.4611556151896573 0 0.0729455299693 ["Character/Dirichlet/693/580"] "1-693-693.587-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.587" [[0, 0.0]] [] 0 true true false false 0.17600595932439395 0 1.07250536634 ["Character/Dirichlet/693/587"] "1-693-693.59-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.59" [[0, 0.0]] [] 0 true true false false 0.4184011346005623 0 1.38997590848 ["Character/Dirichlet/693/59"] "1-693-693.590-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.590" [[0, 0.0]] [] 0 true true false false 0.19346431393885255 0 0.876027452881 ["Character/Dirichlet/693/590"] "1-693-693.592-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.592" [[0, 0.0]] [] 0 true true false false -0.047248051041024294 0 1.07836273639 ["Character/Dirichlet/693/592"] "1-693-693.601-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.601" [[0, 0.0]] [] 0 true true false false 0.4611556151896573 0 1.19262975014 ["Character/Dirichlet/693/601"] "1-693-693.607-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.607" [[0, 0.0]] [] 0 true true false false 0.4374390252289435 0 2.01771626149 ["Character/Dirichlet/693/607"] "1-693-693.608-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.608" [[0, 0.0]] [] 0 true true false false -0.17600595932439395 0 1.00579782349 ["Character/Dirichlet/693/608"] "1-693-693.61-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.61" [[0, 0.0]] [] 0 true true false false 0.18888173532983094 0 0.984014697035 ["Character/Dirichlet/693/61"] "1-693-693.614-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.614" [[0, 0.0]] [] 0 true true false false -0.05733452756799736 0 0.490491147505 ["Character/Dirichlet/693/614"] "1-693-693.625-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.625" [[0, 0.0]] [] 0 true true false false 0.3618137909018107 0 0.0310521961838 ["Character/Dirichlet/693/625"] "1-693-693.65-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.65" [[0, 0.0]] [] 0 true true false false -0.10102463458413806 0 0.394103166439 ["Character/Dirichlet/693/65"] "1-693-693.662-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.662" [[0, 0.0]] [] 0 true true false false -0.05247449824408344 0 0.643890697587 ["Character/Dirichlet/693/662"] "1-693-693.670-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.670" [[0, 0.0]] [] 0 true true false false -0.40906184194283507 0 0.312811368803 ["Character/Dirichlet/693/670"] "1-693-693.74-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.74" [[0, 0.0]] [] 0 true true false false -0.19346431393885255 0 0.777555016042 ["Character/Dirichlet/693/74"] "1-693-693.76-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.76" [[0, 0.0]] [] 0 true true false false -0.2222222222222222 0 0.918261035538 ["Character/Dirichlet/693/76"] "1-693-693.94-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.94" [[0, 0.0]] [] 0 true true false false 0.11811653032661207 0 1.29495934236 ["Character/Dirichlet/693/94"] "1-693-693.95-r0-0-0" 3.218278470074069 3.218278470074069 1 693 "693.95" [[0, 0.0]] [] 0 true true false false 0.3399580275515731 0 1.96055008834 ["Character/Dirichlet/693/95"] "1-693-693.101-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.101" [[1, 0.0]] [] 0 true true false false 0.00556270911461352 0 0.893725731613 ["Character/Dirichlet/693/101"] "1-693-693.103-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.103" [[1, 0.0]] [] 0 true true false false 0.2947218951593367 0 0.666691897877 ["Character/Dirichlet/693/103"] "1-693-693.115-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.115" [[1, 0.0]] [] 0 true true false false -0.2482210279875582 0 0.375952600407 ["Character/Dirichlet/693/115"] "1-693-693.124-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.124" [[1, 0.0]] [] 0 true true false false 0.3315988653994377 0 0.989704041751 ["Character/Dirichlet/693/124"] "1-693-693.131-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.131" [[1, 0.0]] [] 0 true true false false -0.159061841942835 0 0.53006698671 ["Character/Dirichlet/693/131"] "1-693-693.137-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.137" [[1, 0.0]] [] 0 true true false false 0.3881862090981893 0 0.35433119367 ["Character/Dirichlet/693/137"] "1-693-693.142-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.142" [[1, 0.0]] [] 0 true true false false -0.3510246345841381 0 0.711468460005 ["Character/Dirichlet/693/142"] "1-693-693.151-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.151" [[1, 0.0]] [] 0 true true false false -0.4434643139388526 0 1.15957355708 ["Character/Dirichlet/693/151"] "1-693-693.157-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.157" [[1, 0.0]] [] 0 true true false false 0.26083366039621886 0 0.98437205992 ["Character/Dirichlet/693/157"] "1-693-693.158-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.158" [[1, 0.0]] [] 0 true true false false -0.19519162831369022 0 0.179029427351 ["Character/Dirichlet/693/158"] "1-693-693.164-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.164" [[1, 0.0]] [] 0 true true false false 0.159061841942835 0 1.2341151762 ["Character/Dirichlet/693/164"] "1-693-693.167-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.167" [[1, 0.0]] [] 0 true true false false -0.1257213550504437 0 0.975117712681 ["Character/Dirichlet/693/167"] "1-693-693.173-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.173" [[1, 0.0]] [] 0 true true false false -0.45355079046582564 0 1.60656773499 ["Character/Dirichlet/693/173"] "1-693-693.184-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.184" [[1, 0.0]] [] 0 true true false false -0.03440247199601755 0 0.364981545858 ["Character/Dirichlet/693/184"] "1-693-693.191-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.191" [[1, 0.0]] [] 0 true true false false -0.39724089748196634 0 1.99334893378 ["Character/Dirichlet/693/191"] "1-693-693.193-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.193" [[1, 0.0]] [] 0 true true false false 0.4954762325876405 0 0.23243554291 ["Character/Dirichlet/693/193"] "1-693-693.194-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.194" [[1, 0.0]] [] 0 true true false false 0.07987155102460004 0 0.510792865818 ["Character/Dirichlet/693/194"] "1-693-693.202-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.202" [[1, 0.0]] [] 0 true true false false 0.4815615148799496 0 1.48404420245 ["Character/Dirichlet/693/202"] "1-693-693.205-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.205" [[1, 0.0]] [] 0 true true false false -0.11209124161670302 0 0.799557536514 ["Character/Dirichlet/693/205"] "1-693-693.212-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.212" [[1, 0.0]] [] 0 true true false false 0.24925281613075426 0 0.882887587984 ["Character/Dirichlet/693/212"] "1-693-693.223-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.223" [[1, 0.0]] [] 0 true true false false -0.4815615148799496 0 0.0183971749465 ["Character/Dirichlet/693/223"] "1-693-693.227-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.227" [[1, 0.0]] [] 0 true true false false 0.3979952349102701 0 0.804651927429 ["Character/Dirichlet/693/227"] "1-693-693.229-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.229" [[1, 0.0]] [] 0 true true false false 0.2482210279875582 0 1.0515122302 ["Character/Dirichlet/693/229"] "1-693-693.230-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.230" [[1, 0.0]] [] 0 true true false false 0.027777777777777783 0 0.537180042998 ["Character/Dirichlet/693/230"] "1-693-693.248-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.248" [[1, 0.0]] [] 0 true true false false 0.3681165303266121 0 1.27646905958 ["Character/Dirichlet/693/248"] "1-693-693.272-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.272" [[1, 0.0]] [] 0 true true false false -0.26671117074521283 0 0.220278613248 ["Character/Dirichlet/693/272"] "1-693-693.277-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.277" [[1, 0.0]] [] 0 true true false false -0.358030053799639 0 2.05900909263 ["Character/Dirichlet/693/277"] "1-693-693.284-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.284" [[1, 0.0]] [] 0 true true false false -0.24169249548546873 0 0.719490538628 ["Character/Dirichlet/693/284"] "1-693-693.293-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.293" [[1, 0.0]] [] 0 true true false false 0.26671117074521283 0 0.832263116702 ["Character/Dirichlet/693/293"] "1-693-693.299-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.299" [[1, 0.0]] [] 0 true true false false -0.3681165303266121 0 0.364051237076 ["Character/Dirichlet/693/299"] "1-693-693.31-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.31" [[1, 0.0]] [] 0 true true false false -0.3780997325712162 0 0.0818110068201 ["Character/Dirichlet/693/31"] "1-693-693.310-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.310" [[1, 0.0]] [] 0 true true false false 0.05103178814319602 0 0.752231206298 ["Character/Dirichlet/693/310"] "1-693-693.313-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.313" [[1, 0.0]] [] 0 true true false false 0.3780997325712162 0 1.62435310565 ["Character/Dirichlet/693/313"] "1-693-693.317-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.317" [[1, 0.0]] [] 0 true true false false -0.4437417646537448 0 0.0728171854566 ["Character/Dirichlet/693/317"] "1-693-693.328-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.328" [[1, 0.0]] [] 0 true true false false 0.435060647708171 0 0.249961230016 ["Character/Dirichlet/693/328"] "1-693-693.340-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.340" [[1, 0.0]] [] 0 true true false false -0.20453092097141753 0 0.553317833911 ["Character/Dirichlet/693/340"] "1-693-693.346-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.346" [[1, 0.0]] [] 0 true true false false 0.30733452756799734 0 1.39787688031 ["Character/Dirichlet/693/346"] "1-693-693.355-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.355" [[1, 0.0]] [] 0 true true false false -0.11284557904500674 0 0.993368803521 ["Character/Dirichlet/693/355"] "1-693-693.362-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.362" [[1, 0.0]] [] 0 true true false false -0.2146173974983906 0 0.229151032112 ["Character/Dirichlet/693/362"] "1-693-693.367-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.367" [[1, 0.0]] [] 0 true true false false 0.11284557904500674 0 0.708854039105 ["Character/Dirichlet/693/367"] "1-693-693.373-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.373" [[1, 0.0]] [] 0 true true false false 0.20453092097141753 0 0.829290839529 ["Character/Dirichlet/693/373"] "1-693-693.382-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.382" [[1, 0.0]] [] 0 true true false false 0.11209124161670302 0 1.28720191516 ["Character/Dirichlet/693/382"] "1-693-693.389-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.389" [[1, 0.0]] [] 0 true true false false -0.24925281613075426 0 0.667843279654 ["Character/Dirichlet/693/389"] "1-693-693.398-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.398" [[1, 0.0]] [] 0 true true false false -0.18127691060599926 0 0.434380170433 ["Character/Dirichlet/693/398"] "1-693-693.401-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.401" [[1, 0.0]] [] 0 true true false false 0.2972480510410243 0 1.27839409807 ["Character/Dirichlet/693/401"] "1-693-693.403-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.403" [[1, 0.0]] [] 0 true true false false 0.03440247199601755 0 0.760950945046 ["Character/Dirichlet/693/403"] "1-693-693.41-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.41" [[1, 0.0]] [] 0 true true false false -0.21115561518965728 0 0.217411028724 ["Character/Dirichlet/693/41"] "1-693-693.410-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.410" [[1, 0.0]] [] 0 true true false false 0.39724089748196634 0 0.0840051724304 ["Character/Dirichlet/693/410"] "1-693-693.412-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.412" [[1, 0.0]] [] 0 true true false false -0.435060647708171 0 1.69750297209 ["Character/Dirichlet/693/412"] "1-693-693.425-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.425" [[1, 0.0]] [] 0 true true false false 0.024315995469044487 0 0.715328785211 ["Character/Dirichlet/693/425"] "1-693-693.436-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.436" [[1, 0.0]] [] 0 true true false false 0.4434643139388526 0 0.00318219234235 ["Character/Dirichlet/693/436"] "1-693-693.437-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.437" [[1, 0.0]] [] 0 true true false false -0.024315995469044487 0 0.412240929799 ["Character/Dirichlet/693/437"] "1-693-693.443-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.443" [[1, 0.0]] [] 0 true true false false 0.19519162831369022 0 1.093336062 ["Character/Dirichlet/693/443"] "1-693-693.446-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.446" [[1, 0.0]] [] 0 true true false false -0.00556270911461352 0 0.720470314823 ["Character/Dirichlet/693/446"] "1-693-693.454-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.454" [[1, 0.0]] [] 0 true true false false 0.07399404067560605 0 0.48473504502 ["Character/Dirichlet/693/454"] "1-693-693.457-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.457" [[1, 0.0]] [] 0 true true false false -0.19752550175591657 0 0.0739016359313 ["Character/Dirichlet/693/457"] "1-693-693.461-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.461" [[1, 0.0]] [] 0 true true false false -0.027777777777777783 0 0.596542924612 ["Character/Dirichlet/693/461"] "1-693-693.479-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.479" [[1, 0.0]] [] 0 true true false false 0.3125609747710565 0 0.812414229606 ["Character/Dirichlet/693/479"] "1-693-693.493-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.493" [[1, 0.0]] [] 0 true true false false 0.06634471187322823 0 0.552293263831 ["Character/Dirichlet/693/493"] "1-693-693.508-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.508" [[1, 0.0]] [] 0 true true false false 0.19752550175591657 0 0.908855740448 ["Character/Dirichlet/693/508"] "1-693-693.515-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.515" [[1, 0.0]] [] 0 true true false false -0.2972480510410243 0 0.0514822225003 ["Character/Dirichlet/693/515"] "1-693-693.524-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.524" [[1, 0.0]] [] 0 true true false false 0.21115561518965728 0 1.34851225119 ["Character/Dirichlet/693/524"] "1-693-693.544-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.544" [[1, 0.0]] [] 0 true true false false -0.06634471187322823 0 0.184041144847 ["Character/Dirichlet/693/544"] "1-693-693.545-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.545" [[1, 0.0]] [] 0 true true false false 0.18127691060599926 0 1.18569420047 ["Character/Dirichlet/693/545"] "1-693-693.563-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.563" [[1, 0.0]] [] 0 true true false false 0.06111826467016908 0 0.252238768372 ["Character/Dirichlet/693/563"] "1-693-693.565-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.565" [[1, 0.0]] [] 0 true true false false -0.26083366039621886 0 0.241629100464 ["Character/Dirichlet/693/565"] "1-693-693.571-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.571" [[1, 0.0]] [] 0 true true false false 0.3510246345841381 0 1.17941194205 ["Character/Dirichlet/693/571"] "1-693-693.598-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.598" [[1, 0.0]] [] 0 true true false false -0.3315988653994377 0 0.409617834814 ["Character/Dirichlet/693/598"] "1-693-693.599-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.599" [[1, 0.0]] [] 0 true true false false 0.4437417646537448 0 1.3867384547 ["Character/Dirichlet/693/599"] "1-693-693.619-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.619" [[1, 0.0]] [] 0 true true false false -0.2947218951593367 0 0.350585814606 ["Character/Dirichlet/693/619"] "1-693-693.626-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.626" [[1, 0.0]] [] 0 true true false false 0.2146173974983906 0 1.22118218527 ["Character/Dirichlet/693/626"] "1-693-693.632-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.632" [[1, 0.0]] [] 0 true true false false 0.24169249548546873 0 0.758669390528 ["Character/Dirichlet/693/632"] "1-693-693.634-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.634" [[1, 0.0]] [] 0 true true false false -0.4100419724484269 0 0.0847773114537 ["Character/Dirichlet/693/634"] "1-693-693.635-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.635" [[1, 0.0]] [] 0 true true false false -0.3979952349102701 0 0.39430262963 ["Character/Dirichlet/693/635"] "1-693-693.641-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.641" [[1, 0.0]] [] 0 true true false false 0.34168534192641076 0 0.761794673773 ["Character/Dirichlet/693/641"] "1-693-693.643-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.643" [[1, 0.0]] [] 0 true true false false 0.12049490784738455 0 0.958012639524 ["Character/Dirichlet/693/643"] "1-693-693.646-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.646" [[1, 0.0]] [] 0 true true false false 0.4100419724484269 0 1.28625857559 ["Character/Dirichlet/693/646"] "1-693-693.653-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.653" [[1, 0.0]] [] 0 true true false false -0.34168534192641076 0 0.508682005768 ["Character/Dirichlet/693/653"] "1-693-693.655-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.655" [[1, 0.0]] [] 0 true true false false -0.05103178814319602 0 0.909581204951 ["Character/Dirichlet/693/655"] "1-693-693.664-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.664" [[1, 0.0]] [] 0 true true false false -0.07399404067560605 0 0.976728223337 ["Character/Dirichlet/693/664"] "1-693-693.668-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.668" [[1, 0.0]] [] 0 true true false false -0.07987155102460004 0 1.04607117185 ["Character/Dirichlet/693/668"] "1-693-693.677-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.677" [[1, 0.0]] [] 0 true true false false -0.06111826467016908 0 0.637099671146 ["Character/Dirichlet/693/677"] "1-693-693.68-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.68" [[1, 0.0]] [] 0 true true false false -0.3125609747710565 0 0.391391322134 ["Character/Dirichlet/693/68"] "1-693-693.688-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.688" [[1, 0.0]] [] 0 true true false false 0.358030053799639 0 0.124076269897 ["Character/Dirichlet/693/688"] "1-693-693.689-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.689" [[1, 0.0]] [] 0 true true false false 0.45355079046582564 0 0.526491115919 ["Character/Dirichlet/693/689"] "1-693-693.691-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.691" [[1, 0.0]] [] 0 true true false false -0.30733452756799734 0 0.723150348223 ["Character/Dirichlet/693/691"] "1-693-693.79-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.79" [[1, 0.0]] [] 0 true true false false 0.4954762325876405 0 1.08783581923 ["Character/Dirichlet/693/79"] "1-693-693.83-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.83" [[1, 0.0]] [] 0 true true false false 0.1257213550504437 0 1.12636844855 ["Character/Dirichlet/693/83"] "1-693-693.86-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.86" [[1, 0.0]] [] 0 true true false false -0.3881862090981893 0 1.91560724376 ["Character/Dirichlet/693/86"] "1-693-693.97-r1-0-0" 74.47319288267514 74.47319288267514 1 693 "693.97" [[1, 0.0]] [] 0 true true false false -0.12049490784738455 0 0.0589597125826 ["Character/Dirichlet/693/97"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).