# lfunc_search downloaded from the LMFDB on 26 July 2026. # Search link: https://www.lmfdb.org/L/1/837/837.230/r0-0 # Query "{'degree': 1, 'conductor': 837, 'spectral_label': 'r0-0'}" returned 174 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. "1-837-837.103-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.103" [[0, 0.0]] [] 0 true true false false 0.07581228858229618 0 1.35277070713 ["Character/Dirichlet/837/103"] "1-837-837.104-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.104" [[0, 0.0]] [] 0 true true false false 0.4060791604204762 0 2.10533131241 ["Character/Dirichlet/837/104"] "1-837-837.11-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.11" [[0, 0.0]] [] 0 true true false false 0.22089397523529106 0 1.56577468219 ["Character/Dirichlet/837/11"] "1-837-837.110-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.110" [[0, 0.0]] [] 0 true true false false -0.0727458270871429 0 1.02952794942 ["Character/Dirichlet/837/110"] "1-837-837.112-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.112" [[0, 0.0]] [] 0 true true false false -0.03226283459591672 0 0.304886664444 ["Character/Dirichlet/837/112"] "1-837-837.119-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.119" [[0, 0.0]] [] 0 true true false false 0.03508939919375386 0 1.38798592728 ["Character/Dirichlet/837/119"] "1-837-837.121-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.121" [[0, 0.0]] [] 0 true true false false 0.2760395632695557 0 1.28687275962 ["Character/Dirichlet/837/121"] "1-837-837.122-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.122" [[0, 0.0]] [] 0 true true false false -0.39438801116409944 0 1.75528780435 ["Character/Dirichlet/837/122"] "1-837-837.133-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.133" [[0, 0.0]] [] 0 true true false false 0.19038720649979488 0 1.10348144752 ["Character/Dirichlet/837/133"] "1-837-837.137-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.137" [[0, 0.0]] [] 0 true true false false 0.11259157736981866 0 1.48144909265 ["Character/Dirichlet/837/137"] "1-837-837.142-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.142" [[0, 0.0]] [] 0 true true false false 0.03226283459591672 0 0.960699829137 ["Character/Dirichlet/837/142"] "1-837-837.146-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.146" [[0, 0.0]] [] 0 true true false false 0.13290551445984397 0 0.879373214161 ["Character/Dirichlet/837/146"] "1-837-837.157-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.157" [[0, 0.0]] [] 0 true true false false -0.15976590659632522 0 1.22178951483 ["Character/Dirichlet/837/157"] "1-837-837.158-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.158" [[0, 0.0]] [] 0 true true false false 0.4254587543413503 0 0.0355635294073 ["Character/Dirichlet/837/158"] "1-837-837.16-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.16" [[0, 0.0]] [] 0 true true false false 0.15976590659632522 0 1.70179285017 ["Character/Dirichlet/837/16"] "1-837-837.160-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.160" [[0, 0.0]] [] 0 true true false false -0.07310025515243249 0 0.732178675407 ["Character/Dirichlet/837/160"] "1-837-837.167-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.167" [[0, 0.0]] [] 0 true true false false 0.2207417559635147 0 1.43066071927 ["Character/Dirichlet/837/167"] "1-837-837.169-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.169" [[0, 0.0]] [] 0 true true false false 0.4958863287750054 0 1.71959888346 ["Character/Dirichlet/837/169"] "1-837-837.175-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.175" [[0, 0.0]] [] 0 true true false false 0.16255299544167204 0 1.15464235846 ["Character/Dirichlet/837/175"] "1-837-837.176-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.176" [[0, 0.0]] [] 0 true true false false 0.4263930975105016 0 1.86244749545 ["Character/Dirichlet/837/176"] "1-837-837.182-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.182" [[0, 0.0]] [] 0 true true false false -0.13422832379139832 0 1.27747960666 ["Character/Dirichlet/837/182"] "1-837-837.185-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.185" [[0, 0.0]] [] 0 true true false false 0.09259259259259259 0 1.20007889349 ["Character/Dirichlet/837/185"] "1-837-837.193-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.193" [[0, 0.0]] [] 0 true true false false 0.14294612683353847 0 1.27928230912 ["Character/Dirichlet/837/193"] "1-837-837.196-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.196" [[0, 0.0]] [] 0 true true false false 0.390627103397111 0 1.88544266937 ["Character/Dirichlet/837/196"] "1-837-837.202-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.202" [[0, 0.0]] [] 0 true true false false -0.4883822415518229 0 1.38748078147 ["Character/Dirichlet/837/202"] "1-837-837.203-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.203" [[0, 0.0]] [] 0 true true false false 0.4457726914313756 0 1.24813836438 ["Character/Dirichlet/837/203"] "1-837-837.205-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.205" [[0, 0.0]] [] 0 true true false false 0.2825519802188981 0 1.37201365102 ["Character/Dirichlet/837/205"] "1-837-837.209-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.209" [[0, 0.0]] [] 0 true true false false -0.05095686139378686 0 1.27012449779 ["Character/Dirichlet/837/209"] "1-837-837.211-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.211" [[0, 0.0]] [] 0 true true false false 0.40643358848576583 0 0.483949426907 ["Character/Dirichlet/837/211"] "1-837-837.212-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.212" [[0, 0.0]] [] 0 true true false false 0.22027458437893904 0 1.02506985377 ["Character/Dirichlet/837/212"] "1-837-837.214-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.214" [[0, 0.0]] [] 0 true true false false 0.25752104475103715 0 0.887641949395 ["Character/Dirichlet/837/214"] "1-837-837.23-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.23" [[0, 0.0]] [] 0 true true false false 0.13422832379139832 0 0.942218411681 ["Character/Dirichlet/837/23"] "1-837-837.230-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.230" [[0, 0.0]] [] 0 true true false false -0.36888990411166284 0 0.508024700352 ["Character/Dirichlet/837/230"] "1-837-837.239-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.239" [[0, 0.0]] [] 0 true true false false 0.3180906996450292 0 0.980133138108 ["Character/Dirichlet/837/239"] "1-837-837.25-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.25" [[0, 0.0]] [] 0 true true false false 0.054581736633913976 0 0.898150870967 ["Character/Dirichlet/837/25"] "1-837-837.250-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.250" [[0, 0.0]] [] 0 true true false false 0.1550489082184896 0 1.03673574118 ["Character/Dirichlet/837/250"] "1-837-837.254-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.254" [[0, 0.0]] [] 0 true true false false -0.22027458437893904 0 0.739782299968 ["Character/Dirichlet/837/254"] "1-837-837.256-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.256" [[0, 0.0]] [] 0 true true false false 0.26526502887153575 0 1.73445541492 ["Character/Dirichlet/837/256"] "1-837-837.263-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.263" [[0, 0.0]] [] 0 true true false false 0.06105467783076607 0 0.896672814371 ["Character/Dirichlet/837/263"] "1-837-837.268-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.268" [[0, 0.0]] [] 0 true true false false 0.14403447692315352 0 1.7392031056 ["Character/Dirichlet/837/268"] "1-837-837.272-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.272" [[0, 0.0]] [] 0 true true false false 0.015242633688304169 0 1.0682048179 ["Character/Dirichlet/837/272"] "1-837-837.275-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.275" [[0, 0.0]] [] 0 true true false false 0.05095686139378686 0 0.997179481145 ["Character/Dirichlet/837/275"] "1-837-837.283-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.283" [[0, 0.0]] [] 0 true true false false -0.24674651035301723 0 0.879523113362 ["Character/Dirichlet/837/283"] "1-837-837.286-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.286" [[0, 0.0]] [] 0 true true false false 0.45776094164835335 0 0.311032040055 ["Character/Dirichlet/837/286"] "1-837-837.29-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.29" [[0, 0.0]] [] 0 true true false false 0.08709347031738209 0 0.523698371911 ["Character/Dirichlet/837/29"] "1-837-837.290-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.290" [[0, 0.0]] [] 0 true true false false 0.22089397523529106 0 1.29912843918 ["Character/Dirichlet/837/290"] "1-837-837.295-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.295" [[0, 0.0]] [] 0 true true false false 0.49309923992965854 0 0.0584819732556 ["Character/Dirichlet/837/295"] "1-837-837.302-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.302" [[0, 0.0]] [] 0 true true false false 0.13422832379139832 0 1.32241155684 ["Character/Dirichlet/837/302"] "1-837-837.304-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.304" [[0, 0.0]] [] 0 true true false false -0.27875159669941935 0 0.797705292168 ["Character/Dirichlet/837/304"] "1-837-837.308-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.308" [[0, 0.0]] [] 0 true true false false -0.24623986301595124 0 1.0198945587 ["Character/Dirichlet/837/308"] "1-837-837.319-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.319" [[0, 0.0]] [] 0 true true false false -0.45776094164835335 0 1.53726332885 ["Character/Dirichlet/837/319"] "1-837-837.328-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.328" [[0, 0.0]] [] 0 true true false false -0.2825519802188981 0 0.951257623739 ["Character/Dirichlet/837/328"] "1-837-837.344-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.344" [[0, 0.0]] [] 0 true true false false 0.24027356915616513 0 0.965621032544 ["Character/Dirichlet/837/344"] "1-837-837.346-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.346" [[0, 0.0]] [] 0 true true false false 0.27875159669941935 0 1.77137263137 ["Character/Dirichlet/837/346"] "1-837-837.347-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.347" [[0, 0.0]] [] 0 true true false false -0.03508939919375386 0 1.22568692561 ["Character/Dirichlet/837/347"] "1-837-837.349-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.349" [[0, 0.0]] [] 0 true true false false 0.24674651035301723 0 1.25787173286 ["Character/Dirichlet/837/349"] "1-837-837.353-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.353" [[0, 0.0]] [] 0 true true false false 0.36888990411166284 0 1.23370250331 ["Character/Dirichlet/837/353"] "1-837-837.355-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.355" [[0, 0.0]] [] 0 true true false false 0.18929885641017982 0 1.42116597999 ["Character/Dirichlet/837/355"] "1-837-837.356-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.356" [[0, 0.0]] [] 0 true true false false 0.24623986301595124 0 1.4226664193 ["Character/Dirichlet/837/356"] "1-837-837.362-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.362" [[0, 0.0]] [] 0 true true false false -0.4254587543413503 0 0.590282573242 ["Character/Dirichlet/837/362"] "1-837-837.365-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.365" [[0, 0.0]] [] 0 true true false false -0.46623884779317737 0 1.49641976288 ["Character/Dirichlet/837/365"] "1-837-837.371-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.371" [[0, 0.0]] [] 0 true true false false -0.09259259259259259 0 0.553002824645 ["Character/Dirichlet/837/371"] "1-837-837.376-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.376" [[0, 0.0]] [] 0 true true false false -0.26526502887153575 0 1.05110020877 ["Character/Dirichlet/837/376"] "1-837-837.382-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.382" [[0, 0.0]] [] 0 true true false false 0.07581228858229618 0 1.12807482125 ["Character/Dirichlet/837/382"] "1-837-837.383-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.383" [[0, 0.0]] [] 0 true true false false 0.4060791604204762 0 0.260141553322 ["Character/Dirichlet/837/383"] "1-837-837.389-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.389" [[0, 0.0]] [] 0 true true false false -0.4060791604204762 0 1.61532667112 ["Character/Dirichlet/837/389"] "1-837-837.391-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.391" [[0, 0.0]] [] 0 true true false false -0.36559616792925004 0 0.793580251451 ["Character/Dirichlet/837/391"] "1-837-837.398-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.398" [[0, 0.0]] [] 0 true true false false 0.03508939919375386 0 0.832455753145 ["Character/Dirichlet/837/398"] "1-837-837.4-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.4" [[0, 0.0]] [] 0 true true false false 0.08658682298031613 0 0.704939200059 ["Character/Dirichlet/837/4"] "1-837-837.40-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.40" [[0, 0.0]] [] 0 true true false false 0.2089057250183134 0 1.51481355856 ["Character/Dirichlet/837/40"] "1-837-837.400-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.400" [[0, 0.0]] [] 0 true true false false -0.390627103397111 0 0.0979287807599 ["Character/Dirichlet/837/400"] "1-837-837.401-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.401" [[0, 0.0]] [] 0 true true false false -0.06105467783076607 0 1.37330002637 ["Character/Dirichlet/837/401"] "1-837-837.412-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.412" [[0, 0.0]] [] 0 true true false false -0.14294612683353847 0 0.85192895517 ["Character/Dirichlet/837/412"] "1-837-837.416-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.416" [[0, 0.0]] [] 0 true true false false -0.2207417559635147 0 0.851698084494 ["Character/Dirichlet/837/416"] "1-837-837.421-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.421" [[0, 0.0]] [] 0 true true false false 0.03226283459591672 0 1.12697239371 ["Character/Dirichlet/837/421"] "1-837-837.425-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.425" [[0, 0.0]] [] 0 true true false false 0.13290551445984397 0 1.66362862747 ["Character/Dirichlet/837/425"] "1-837-837.436-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.436" [[0, 0.0]] [] 0 true true false false -0.15976590659632522 0 0.309190953185 ["Character/Dirichlet/837/436"] "1-837-837.437-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.437" [[0, 0.0]] [] 0 true true false false 0.4254587543413503 0 2.25362579877 ["Character/Dirichlet/837/437"] "1-837-837.439-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.439" [[0, 0.0]] [] 0 true true false false -0.40643358848576583 0 0.903628195971 ["Character/Dirichlet/837/439"] "1-837-837.446-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.446" [[0, 0.0]] [] 0 true true false false -0.11259157736981866 0 0.352767434262 ["Character/Dirichlet/837/446"] "1-837-837.448-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.448" [[0, 0.0]] [] 0 true true false false 0.4958863287750054 0 0.614362457688 ["Character/Dirichlet/837/448"] "1-837-837.454-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.454" [[0, 0.0]] [] 0 true true false false 0.4958863287750054 0 1.74074537732 ["Character/Dirichlet/837/454"] "1-837-837.455-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.455" [[0, 0.0]] [] 0 true true false false -0.24027356915616513 0 0.77712804265 ["Character/Dirichlet/837/455"] "1-837-837.461-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.461" [[0, 0.0]] [] 0 true true false false 0.19910500954193502 0 0.971906806898 ["Character/Dirichlet/837/461"] "1-837-837.464-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.464" [[0, 0.0]] [] 0 true true false false 0.09259259259259259 0 0.462837056517 ["Character/Dirichlet/837/464"] "1-837-837.472-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.472" [[0, 0.0]] [] 0 true true false false -0.19038720649979488 0 1.15293156457 ["Character/Dirichlet/837/472"] "1-837-837.475-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.475" [[0, 0.0]] [] 0 true true false false 0.390627103397111 0 0.949797705813 ["Character/Dirichlet/837/475"] "1-837-837.481-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.481" [[0, 0.0]] [] 0 true true false false -0.1550489082184896 0 0.925147247351 ["Character/Dirichlet/837/481"] "1-837-837.482-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.482" [[0, 0.0]] [] 0 true true false false -0.22089397523529106 0 0.0996753942919 ["Character/Dirichlet/837/482"] "1-837-837.484-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.484" [[0, 0.0]] [] 0 true true false false -0.3841146864477686 0 0.402554737488 ["Character/Dirichlet/837/484"] "1-837-837.488-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.488" [[0, 0.0]] [] 0 true true false false 0.2823764719395465 0 1.60240448962 ["Character/Dirichlet/837/488"] "1-837-837.49-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.49" [[0, 0.0]] [] 0 true true false false -0.2825519802188981 0 0.502538970258 ["Character/Dirichlet/837/49"] "1-837-837.490-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.490" [[0, 0.0]] [] 0 true true false false 0.40643358848576583 0 1.89952752665 ["Character/Dirichlet/837/490"] "1-837-837.491-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.491" [[0, 0.0]] [] 0 true true false false 0.22027458437893904 0 1.01371552486 ["Character/Dirichlet/837/491"] "1-837-837.493-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.493" [[0, 0.0]] [] 0 true true false false -0.07581228858229618 0 1.14547903353 ["Character/Dirichlet/837/493"] "1-837-837.509-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.509" [[0, 0.0]] [] 0 true true false false -0.035556570778329494 0 0.564577280828 ["Character/Dirichlet/837/509"] "1-837-837.518-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.518" [[0, 0.0]] [] 0 true true false false 0.3180906996450292 0 0.109030546864 ["Character/Dirichlet/837/518"] "1-837-837.529-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.529" [[0, 0.0]] [] 0 true true false false 0.1550489082184896 0 1.14194537324 ["Character/Dirichlet/837/529"] "1-837-837.533-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.533" [[0, 0.0]] [] 0 true true false false -0.22027458437893904 0 0.308186718858 ["Character/Dirichlet/837/533"] "1-837-837.535-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.535" [[0, 0.0]] [] 0 true true false false -0.401401637795131 0 0.1513997279 ["Character/Dirichlet/837/535"] "1-837-837.542-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.542" [[0, 0.0]] [] 0 true true false false 0.39438801116409944 0 0.0764528950005 ["Character/Dirichlet/837/542"] "1-837-837.547-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.547" [[0, 0.0]] [] 0 true true false false -0.18929885641017982 0 0.544907343072 ["Character/Dirichlet/837/547"] "1-837-837.551-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.551" [[0, 0.0]] [] 0 true true false false -0.3180906996450292 0 2.2676394632 ["Character/Dirichlet/837/551"] "1-837-837.554-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.554" [[0, 0.0]] [] 0 true true false false -0.2823764719395465 0 0.468467984848 ["Character/Dirichlet/837/554"] "1-837-837.562-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.562" [[0, 0.0]] [] 0 true true false false 0.08658682298031613 0 0.895525591665 ["Character/Dirichlet/837/562"] "1-837-837.565-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.565" [[0, 0.0]] [] 0 true true false false -0.2089057250183134 0 0.82272916586 ["Character/Dirichlet/837/565"] "1-837-837.569-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.569" [[0, 0.0]] [] 0 true true false false -0.4457726914313756 0 0.00327532191682 ["Character/Dirichlet/837/569"] "1-837-837.574-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.574" [[0, 0.0]] [] 0 true true false false 0.15976590659632522 0 1.20337271338 ["Character/Dirichlet/837/574"] "1-837-837.581-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.581" [[0, 0.0]] [] 0 true true false false -0.19910500954193502 0 0.575428062958 ["Character/Dirichlet/837/581"] "1-837-837.583-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.583" [[0, 0.0]] [] 0 true true false false -0.27875159669941935 0 0.245070946419 ["Character/Dirichlet/837/583"] "1-837-837.587-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.587" [[0, 0.0]] [] 0 true true false false -0.24623986301595124 0 0.741963920966 ["Character/Dirichlet/837/587"] "1-837-837.598-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.598" [[0, 0.0]] [] 0 true true false false -0.45776094164835335 0 0.600316002937 ["Character/Dirichlet/837/598"] "1-837-837.607-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.607" [[0, 0.0]] [] 0 true true false false 0.3841146864477686 0 0.965100482448 ["Character/Dirichlet/837/607"] "1-837-837.623-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.623" [[0, 0.0]] [] 0 true true false false -0.4263930975105016 0 0.791463181059 ["Character/Dirichlet/837/623"] "1-837-837.625-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.625" [[0, 0.0]] [] 0 true true false false 0.27875159669941935 0 1.44906802675 ["Character/Dirichlet/837/625"] "1-837-837.626-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.626" [[0, 0.0]] [] 0 true true false false -0.03508939919375386 0 0.636007448315 ["Character/Dirichlet/837/626"] "1-837-837.628-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.628" [[0, 0.0]] [] 0 true true false false -0.08658682298031613 0 0.394433448651 ["Character/Dirichlet/837/628"] "1-837-837.632-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.632" [[0, 0.0]] [] 0 true true false false 0.035556570778329494 0 1.35897360111 ["Character/Dirichlet/837/632"] "1-837-837.634-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.634" [[0, 0.0]] [] 0 true true false false -0.14403447692315352 0 0.827873594146 ["Character/Dirichlet/837/634"] "1-837-837.635-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.635" [[0, 0.0]] [] 0 true true false false -0.08709347031738209 0 1.08679922852 ["Character/Dirichlet/837/635"] "1-837-837.641-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.641" [[0, 0.0]] [] 0 true true false false -0.4254587543413503 0 1.92699600628 ["Character/Dirichlet/837/641"] "1-837-837.644-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.644" [[0, 0.0]] [] 0 true true false false -0.13290551445984397 0 1.13013830564 ["Character/Dirichlet/837/644"] "1-837-837.65-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.65" [[0, 0.0]] [] 0 true true false false 0.24027356915616513 0 1.1483529832 ["Character/Dirichlet/837/65"] "1-837-837.650-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.650" [[0, 0.0]] [] 0 true true false false 0.24074074074074073 0 1.64536003935 ["Character/Dirichlet/837/650"] "1-837-837.655-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.655" [[0, 0.0]] [] 0 true true false false 0.401401637795131 0 1.12139625604 ["Character/Dirichlet/837/655"] "1-837-837.661-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.661" [[0, 0.0]] [] 0 true true false false -0.25752104475103715 0 0.488751477298 ["Character/Dirichlet/837/661"] "1-837-837.662-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.662" [[0, 0.0]] [] 0 true true false false 0.0727458270871429 0 0.844832797789 ["Character/Dirichlet/837/662"] "1-837-837.668-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.668" [[0, 0.0]] [] 0 true true false false -0.4060791604204762 0 0.417121454054 ["Character/Dirichlet/837/668"] "1-837-837.67-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.67" [[0, 0.0]] [] 0 true true false false -0.054581736633913976 0 0.786757673623 ["Character/Dirichlet/837/67"] "1-837-837.670-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.670" [[0, 0.0]] [] 0 true true false false -0.03226283459591672 0 1.12880408857 ["Character/Dirichlet/837/670"] "1-837-837.677-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.677" [[0, 0.0]] [] 0 true true false false 0.3684227325270872 0 1.55670308974 ["Character/Dirichlet/837/677"] "1-837-837.679-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.679" [[0, 0.0]] [] 0 true true false false -0.390627103397111 0 0.548001872908 ["Character/Dirichlet/837/679"] "1-837-837.68-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.68" [[0, 0.0]] [] 0 true true false false -0.3684227325270872 0 0.358710616846 ["Character/Dirichlet/837/68"] "1-837-837.680-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.680" [[0, 0.0]] [] 0 true true false false -0.06105467783076607 0 0.540316058195 ["Character/Dirichlet/837/680"] "1-837-837.691-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.691" [[0, 0.0]] [] 0 true true false false -0.14294612683353847 0 0.673860250516 ["Character/Dirichlet/837/691"] "1-837-837.695-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.695" [[0, 0.0]] [] 0 true true false false -0.2207417559635147 0 2.20802862809 ["Character/Dirichlet/837/695"] "1-837-837.7-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.7" [[0, 0.0]] [] 0 true true false false 0.45776094164835335 0 1.44401349566 ["Character/Dirichlet/837/7"] "1-837-837.70-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.70" [[0, 0.0]] [] 0 true true false false -0.08658682298031613 0 1.41748663789 ["Character/Dirichlet/837/70"] "1-837-837.700-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.700" [[0, 0.0]] [] 0 true true false false 0.36559616792925004 0 1.5864339888 ["Character/Dirichlet/837/700"] "1-837-837.704-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.704" [[0, 0.0]] [] 0 true true false false 0.46623884779317737 0 0.545725293931 ["Character/Dirichlet/837/704"] "1-837-837.715-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.715" [[0, 0.0]] [] 0 true true false false -0.49309923992965854 0 1.9591663133 ["Character/Dirichlet/837/715"] "1-837-837.716-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.716" [[0, 0.0]] [] 0 true true false false 0.09212542100801696 0 1.33288312704 ["Character/Dirichlet/837/716"] "1-837-837.718-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.718" [[0, 0.0]] [] 0 true true false false -0.40643358848576583 0 1.62835854755 ["Character/Dirichlet/837/718"] "1-837-837.725-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.725" [[0, 0.0]] [] 0 true true false false 0.2207417559635147 0 0.140688005477 ["Character/Dirichlet/837/725"] "1-837-837.727-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.727" [[0, 0.0]] [] 0 true true false false -0.16255299544167204 0 0.0421965004504 ["Character/Dirichlet/837/727"] "1-837-837.733-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.733" [[0, 0.0]] [] 0 true true false false 0.4958863287750054 0 0.0541657054372 ["Character/Dirichlet/837/733"] "1-837-837.734-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.734" [[0, 0.0]] [] 0 true true false false -0.24027356915616513 0 0.637102937062 ["Character/Dirichlet/837/734"] "1-837-837.74-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.74" [[0, 0.0]] [] 0 true true false false 0.035556570778329494 0 1.09449875634 ["Character/Dirichlet/837/74"] "1-837-837.740-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.740" [[0, 0.0]] [] 0 true true false false -0.13422832379139832 0 0.352329123899 ["Character/Dirichlet/837/740"] "1-837-837.743-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.743" [[0, 0.0]] [] 0 true true false false -0.24074074074074073 0 1.0327093331 ["Character/Dirichlet/837/743"] "1-837-837.751-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.751" [[0, 0.0]] [] 0 true true false false 0.14294612683353847 0 0.608742529678 ["Character/Dirichlet/837/751"] "1-837-837.754-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.754" [[0, 0.0]] [] 0 true true false false -0.2760395632695557 0 0.813036955021 ["Character/Dirichlet/837/754"] "1-837-837.76-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.76" [[0, 0.0]] [] 0 true true false false 0.18929885641017982 0 1.04796401892 ["Character/Dirichlet/837/76"] "1-837-837.760-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.760" [[0, 0.0]] [] 0 true true false false -0.1550489082184896 0 0.969825287648 ["Character/Dirichlet/837/760"] "1-837-837.761-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.761" [[0, 0.0]] [] 0 true true false false -0.22089397523529106 0 1.2410816631 ["Character/Dirichlet/837/761"] "1-837-837.763-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.763" [[0, 0.0]] [] 0 true true false false 0.2825519802188981 0 1.69980457701 ["Character/Dirichlet/837/763"] "1-837-837.767-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.767" [[0, 0.0]] [] 0 true true false false -0.05095686139378686 0 0.653724633923 ["Character/Dirichlet/837/767"] "1-837-837.769-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.769" [[0, 0.0]] [] 0 true true false false 0.07310025515243249 0 0.881869999852 ["Character/Dirichlet/837/769"] "1-837-837.77-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.77" [[0, 0.0]] [] 0 true true false false 0.24623986301595124 0 1.19959501318 ["Character/Dirichlet/837/77"] "1-837-837.770-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.770" [[0, 0.0]] [] 0 true true false false -0.1130587489543943 0 1.34212778698 ["Character/Dirichlet/837/770"] "1-837-837.772-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.772" [[0, 0.0]] [] 0 true true false false -0.07581228858229618 0 1.00671241221 ["Character/Dirichlet/837/772"] "1-837-837.788-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.788" [[0, 0.0]] [] 0 true true false false -0.035556570778329494 0 0.927846383022 ["Character/Dirichlet/837/788"] "1-837-837.797-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.797" [[0, 0.0]] [] 0 true true false false -0.015242633688304169 0 1.04942473605 ["Character/Dirichlet/837/797"] "1-837-837.808-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.808" [[0, 0.0]] [] 0 true true false false 0.4883822415518229 0 0.393015287836 ["Character/Dirichlet/837/808"] "1-837-837.812-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.812" [[0, 0.0]] [] 0 true true false false 0.1130587489543943 0 1.44586772754 ["Character/Dirichlet/837/812"] "1-837-837.814-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.814" [[0, 0.0]] [] 0 true true false false -0.401401637795131 0 0.485602157651 ["Character/Dirichlet/837/814"] "1-837-837.821-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.821" [[0, 0.0]] [] 0 true true false false 0.06105467783076607 0 0.792295169583 ["Character/Dirichlet/837/821"] "1-837-837.826-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.826" [[0, 0.0]] [] 0 true true false false -0.18929885641017982 0 0.670845314247 ["Character/Dirichlet/837/826"] "1-837-837.83-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.83" [[0, 0.0]] [] 0 true true false false -0.09212542100801696 0 0.533178287412 ["Character/Dirichlet/837/83"] "1-837-837.830-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.830" [[0, 0.0]] [] 0 true true false false -0.3180906996450292 0 0.678756526337 ["Character/Dirichlet/837/830"] "1-837-837.833-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.833" [[0, 0.0]] [] 0 true true false false 0.05095686139378686 0 1.3911407059 ["Character/Dirichlet/837/833"] "1-837-837.86-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.86" [[0, 0.0]] [] 0 true true false false -0.13290551445984397 0 0.901242078152 ["Character/Dirichlet/837/86"] "1-837-837.92-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.92" [[0, 0.0]] [] 0 true true false false -0.09259259259259259 0 0.815873033123 ["Character/Dirichlet/837/92"] "1-837-837.97-r0-0-0" 3.8870116586608887 3.8870116586608887 1 837 "837.97" [[0, 0.0]] [] 0 true true false false 0.401401637795131 0 1.65208824585 ["Character/Dirichlet/837/97"] # 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).