# lfunc_search downloaded from the LMFDB on 27 July 2026. # Search link: https://www.lmfdb.org/L/1/837/837.275 # Query "{'degree': 1, 'conductor': 837}" returned 348 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 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"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"] "1-837-837.101-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.101" [[1, 0.0]] [] 0 true true false false 0.1634131770196839 0 1.27925958061 ["Character/Dirichlet/837/101"] "1-837-837.106-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.106" [[1, 0.0]] [] 0 true true false false 0.3625915773698186 0 1.01759321118 ["Character/Dirichlet/837/106"] "1-837-837.113-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.113" [[1, 0.0]] [] 0 true true false false 0.24588632877500538 0 0.0161899159305 ["Character/Dirichlet/837/113"] "1-837-837.115-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.115" [[1, 0.0]] [] 0 true true false false 0.382905514459844 0 1.1829089409 ["Character/Dirichlet/837/115"] "1-837-837.128-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.128" [[1, 0.0]] [] 0 true true false false -0.34859836220486906 0 0.805063438653 ["Character/Dirichlet/837/128"] "1-837-837.13-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.13" [[1, 0.0]] [] 0 true true false false 0.38111009588833716 0 0.631249053767 ["Character/Dirichlet/837/13"] "1-837-837.130-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.130" [[1, 0.0]] [] 0 true true false false -0.28508939919375387 0 0.42597621609 ["Character/Dirichlet/837/130"] "1-837-837.131-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.131" [[1, 0.0]] [] 0 true true false false -0.10705387316646152 0 0.894460814041 ["Character/Dirichlet/837/131"] "1-837-837.139-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.139" [[1, 0.0]] [] 0 true true false false 0.003760136984048749 0 0.679864373305 ["Character/Dirichlet/837/139"] "1-837-837.14-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.14" [[1, 0.0]] [] 0 true true false false -0.3940344769231536 0 0.00331340970288 ["Character/Dirichlet/837/14"] "1-837-837.140-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.140" [[1, 0.0]] [] 0 true true false false -0.40504890821848966 0 1.84946006335 ["Character/Dirichlet/837/140"] "1-837-837.148-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.148" [[1, 0.0]] [] 0 true true false false 0.2837611522068227 0 0.0565915059634 ["Character/Dirichlet/837/148"] "1-837-837.149-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.149" [[1, 0.0]] [] 0 true true false false -0.17689974484756754 0 0.519292766317 ["Character/Dirichlet/837/149"] "1-837-837.151-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.151" [[1, 0.0]] [] 0 true true false false 0.11577167620860168 0 1.02590949843 ["Character/Dirichlet/837/151"] "1-837-837.164-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.164" [[1, 0.0]] [] 0 true true false false 0.4403872064997949 0 0.130395576836 ["Character/Dirichlet/837/164"] "1-837-837.166-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.166" [[1, 0.0]] [] 0 true true false false 0.15607916042047623 0 1.02630377369 ["Character/Dirichlet/837/166"] "1-837-837.173-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.173" [[1, 0.0]] [] 0 true true false false 0.2822628345959167 0 1.27316293629 ["Character/Dirichlet/837/173"] "1-837-837.178-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.178" [[1, 0.0]] [] 0 true true false false 0.19904313860621314 0 1.08828475279 ["Character/Dirichlet/837/178"] "1-837-837.184-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.184" [[1, 0.0]] [] 0 true true false false -0.31105467783076607 0 0.707491297258 ["Character/Dirichlet/837/184"] "1-837-837.191-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.191" [[1, 0.0]] [] 0 true true false false 0.17689974484756754 0 0.882952837145 ["Character/Dirichlet/837/191"] "1-837-837.194-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.194" [[1, 0.0]] [] 0 true true false false 0.34859836220486906 0 0.267170223807 ["Character/Dirichlet/837/194"] "1-837-837.2-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.2" [[1, 0.0]] [] 0 true true false false -0.2616177584481771 0 1.54755697328 ["Character/Dirichlet/837/2"] "1-837-837.20-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.20" [[1, 0.0]] [] 0 true true false false 0.06070114358982019 0 0.877348083743 ["Character/Dirichlet/837/20"] "1-837-837.200-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.200" [[1, 0.0]] [] 0 true true false false -0.24588632877500538 0 1.48935904501 ["Character/Dirichlet/837/200"] "1-837-837.22-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.22" [[1, 0.0]] [] 0 true true false false 0.06809069964502917 0 0.363566047226 ["Character/Dirichlet/837/22"] "1-837-837.220-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.220" [[1, 0.0]] [] 0 true true false false 0.1754587543413503 0 1.29930265403 ["Character/Dirichlet/837/220"] "1-837-837.221-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.221" [[1, 0.0]] [] 0 true true false false -0.1634131770196839 0 0.299353113182 ["Character/Dirichlet/837/221"] "1-837-837.223-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.223" [[1, 0.0]] [] 0 true true false false 0.02972541562106096 0 1.06543337713 ["Character/Dirichlet/837/223"] "1-837-837.227-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.227" [[1, 0.0]] [] 0 true true false false -0.359372896602889 0 0.689193968177 ["Character/Dirichlet/837/227"] "1-837-837.229-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.229" [[1, 0.0]] [] 0 true true false false -0.3625915773698186 0 0.523962304798 ["Character/Dirichlet/837/229"] "1-837-837.232-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.232" [[1, 0.0]] [] 0 true true false false 0.31105467783076607 0 1.803929675 ["Character/Dirichlet/837/232"] "1-837-837.236-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.236" [[1, 0.0]] [] 0 true true false false -0.4674480197811019 0 0.267180685641 ["Character/Dirichlet/837/236"] "1-837-837.238-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.238" [[1, 0.0]] [] 0 true true false false -0.4902735691561651 0 1.77965455131 ["Character/Dirichlet/837/238"] "1-837-837.241-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.241" [[1, 0.0]] [] 0 true true false false 0.2652426336883042 0 1.48389076913 ["Character/Dirichlet/837/241"] "1-837-837.245-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.245" [[1, 0.0]] [] 0 true true false false -0.49247895524896285 0 0.565960354689 ["Character/Dirichlet/837/245"] "1-837-837.247-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.247" [[1, 0.0]] [] 0 true true false false -0.1574074074074074 0 0.279687007806 ["Character/Dirichlet/837/247"] "1-837-837.257-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.257" [[1, 0.0]] [] 0 true true false false -0.04109427498168663 0 0.76422825472 ["Character/Dirichlet/837/257"] "1-837-837.259-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.259" [[1, 0.0]] [] 0 true true false false 0.4708939752352911 0 0.00362264349783 ["Character/Dirichlet/837/259"] "1-837-837.265-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.265" [[1, 0.0]] [] 0 true true false false -0.4708939752352911 0 1.01669272556 ["Character/Dirichlet/837/265"] "1-837-837.266-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.266" [[1, 0.0]] [] 0 true true false false 0.4674480197811019 0 1.09743415033 ["Character/Dirichlet/837/266"] "1-837-837.274-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.274" [[1, 0.0]] [] 0 true true false false -0.02972541562106096 0 0.50473639874 ["Character/Dirichlet/837/274"] "1-837-837.277-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.277" [[1, 0.0]] [] 0 true true false false 0.003760136984048749 0 0.30978619343 ["Character/Dirichlet/837/277"] "1-837-837.281-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.281" [[1, 0.0]] [] 0 true true false false 0.40504890821848966 0 0.332079083215 ["Character/Dirichlet/837/281"] "1-837-837.284-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.284" [[1, 0.0]] [] 0 true true false false -0.30458173663391397 0 0.750819405555 ["Character/Dirichlet/837/284"] "1-837-837.292-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.292" [[1, 0.0]] [] 0 true true false false -0.28555657077832947 0 0.20606217013 ["Character/Dirichlet/837/292"] "1-837-837.293-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.293" [[1, 0.0]] [] 0 true true false false -0.06070114358982019 0 0.317741891374 ["Character/Dirichlet/837/293"] "1-837-837.299-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.299" [[1, 0.0]] [] 0 true true false false 0.3940344769231536 0 0.772453030624 ["Character/Dirichlet/837/299"] "1-837-837.301-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.301" [[1, 0.0]] [] 0 true true false false 0.06809069964502917 0 0.849229555897 ["Character/Dirichlet/837/301"] "1-837-837.313-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.313" [[1, 0.0]] [] 0 true true false false 0.4902735691561651 0 0.062922412747 ["Character/Dirichlet/837/313"] "1-837-837.317-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.317" [[1, 0.0]] [] 0 true true false false -0.2922390583516467 0 0.141578364597 ["Character/Dirichlet/837/317"] "1-837-837.320-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.320" [[1, 0.0]] [] 0 true true false false -0.17418771141770384 0 0.66067901103 ["Character/Dirichlet/837/320"] "1-837-837.322-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.322" [[1, 0.0]] [] 0 true true false false -0.38111009588833716 0 0.300431896426 ["Character/Dirichlet/837/322"] "1-837-837.326-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.326" [[1, 0.0]] [] 0 true true false false -0.25690076007034146 0 0.343980601358 ["Character/Dirichlet/837/326"] "1-837-837.329-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.329" [[1, 0.0]] [] 0 true true false false -0.2822628345959167 0 0.267640380853 ["Character/Dirichlet/837/329"] "1-837-837.331-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.331" [[1, 0.0]] [] 0 true true false false -0.1754587543413503 0 0.373237849831 ["Character/Dirichlet/837/331"] "1-837-837.335-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.335" [[1, 0.0]] [] 0 true true false false -0.028751596699419367 0 0.870500103718 ["Character/Dirichlet/837/335"] "1-837-837.337-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.337" [[1, 0.0]] [] 0 true true false false 0.46762352806045354 0 0.223721250914 ["Character/Dirichlet/837/337"] "1-837-837.338-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.338" [[1, 0.0]] [] 0 true true false false 0.026039563269555675 0 0.70345744104 ["Character/Dirichlet/837/338"] "1-837-837.34-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.34" [[1, 0.0]] [] 0 true true false false 0.4902735691561651 0 1.66126321089 ["Character/Dirichlet/837/34"] "1-837-837.340-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.340" [[1, 0.0]] [] 0 true true false false 0.1574074074074074 0 0.476742445623 ["Character/Dirichlet/837/340"] "1-837-837.358-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.358" [[1, 0.0]] [] 0 true true false false -0.15607916042047623 0 0.611236500914 ["Character/Dirichlet/837/358"] "1-837-837.364-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.364" [[1, 0.0]] [] 0 true true false false -0.449105009541935 0 1.89309321006 ["Character/Dirichlet/837/364"] "1-837-837.367-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.367" [[1, 0.0]] [] 0 true true false false 0.28508939919375387 0 0.497574501824 ["Character/Dirichlet/837/367"] "1-837-837.374-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.374" [[1, 0.0]] [] 0 true true false false -0.4097659065963252 0 0.345306134049 ["Character/Dirichlet/837/374"] "1-837-837.38-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.38" [[1, 0.0]] [] 0 true true false false -0.2922390583516467 0 1.68856330739 ["Character/Dirichlet/837/38"] "1-837-837.380-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.380" [[1, 0.0]] [] 0 true true false false 0.49674651035301726 0 0.0461803587513 ["Character/Dirichlet/837/380"] "1-837-837.385-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.385" [[1, 0.0]] [] 0 true true false false 0.02925824403648532 0 0.969432161156 ["Character/Dirichlet/837/385"] "1-837-837.392-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.392" [[1, 0.0]] [] 0 true true false false -0.08744700455832798 0 0.231933752322 ["Character/Dirichlet/837/392"] "1-837-837.394-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.394" [[1, 0.0]] [] 0 true true false false 0.382905514459844 0 0.0806550637241 ["Character/Dirichlet/837/394"] "1-837-837.407-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.407" [[1, 0.0]] [] 0 true true false false -0.015265028871535724 0 0.476939114784 ["Character/Dirichlet/837/407"] "1-837-837.409-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.409" [[1, 0.0]] [] 0 true true false false -0.28508939919375387 0 2.06616665795 ["Character/Dirichlet/837/409"] "1-837-837.41-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.41" [[1, 0.0]] [] 0 true true false false 0.49247895524896285 0 1.52135984768 ["Character/Dirichlet/837/41"] "1-837-837.410-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.410" [[1, 0.0]] [] 0 true true false false -0.10705387316646152 0 0.689217441092 ["Character/Dirichlet/837/410"] "1-837-837.418-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.418" [[1, 0.0]] [] 0 true true false false -0.3370934703173821 0 1.23796533065 ["Character/Dirichlet/837/418"] "1-837-837.419-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.419" [[1, 0.0]] [] 0 true true false false 0.2616177584481771 0 0.424614245049 ["Character/Dirichlet/837/419"] "1-837-837.427-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.427" [[1, 0.0]] [] 0 true true false false -0.382905514459844 0 1.11817552462 ["Character/Dirichlet/837/427"] "1-837-837.428-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.428" [[1, 0.0]] [] 0 true true false false 0.1564335884857658 0 0.751131556752 ["Character/Dirichlet/837/428"] "1-837-837.43-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.43" [[1, 0.0]] [] 0 true true false false 0.28555657077832947 0 0.794923036316 ["Character/Dirichlet/837/43"] "1-837-837.430-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.430" [[1, 0.0]] [] 0 true true false false 0.449105009541935 0 0.362051376017 ["Character/Dirichlet/837/430"] "1-837-837.443-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.443" [[1, 0.0]] [] 0 true true false false 0.10705387316646152 0 0.633495204044 ["Character/Dirichlet/837/443"] "1-837-837.445-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.445" [[1, 0.0]] [] 0 true true false false -0.17725417291285708 0 0.546630093698 ["Character/Dirichlet/837/445"] "1-837-837.452-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.452" [[1, 0.0]] [] 0 true true false false 0.2822628345959167 0 0.635384659937 ["Character/Dirichlet/837/452"] "1-837-837.457-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.457" [[1, 0.0]] [] 0 true true false false -0.46762352806045354 0 1.39781406802 ["Character/Dirichlet/837/457"] "1-837-837.463-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.463" [[1, 0.0]] [] 0 true true false false -0.31105467783076607 0 1.27128889821 ["Character/Dirichlet/837/463"] "1-837-837.47-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.47" [[1, 0.0]] [] 0 true true false false 0.4097659065963252 0 1.09904566422 ["Character/Dirichlet/837/47"] "1-837-837.470-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.470" [[1, 0.0]] [] 0 true true false false -0.1564335884857658 0 0.752649782277 ["Character/Dirichlet/837/470"] "1-837-837.473-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.473" [[1, 0.0]] [] 0 true true false false 0.015265028871535724 0 0.682264017915 ["Character/Dirichlet/837/473"] "1-837-837.479-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.479" [[1, 0.0]] [] 0 true true false false -0.24588632877500538 0 0.736540660043 ["Character/Dirichlet/837/479"] "1-837-837.499-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.499" [[1, 0.0]] [] 0 true true false false -0.15787457899198307 0 0.37612081639 ["Character/Dirichlet/837/499"] "1-837-837.5-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.5" [[1, 0.0]] [] 0 true true false false 0.028751596699419367 0 0.652196691974 ["Character/Dirichlet/837/5"] "1-837-837.50-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.50" [[1, 0.0]] [] 0 true true false false -0.2822628345959167 0 0.6431248324 ["Character/Dirichlet/837/50"] "1-837-837.500-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.500" [[1, 0.0]] [] 0 true true false false 0.49674651035301726 0 1.2615406577 ["Character/Dirichlet/837/500"] "1-837-837.502-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.502" [[1, 0.0]] [] 0 true true false false 0.02972541562106096 0 0.798216998577 ["Character/Dirichlet/837/502"] "1-837-837.506-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.506" [[1, 0.0]] [] 0 true true false false -0.359372896602889 0 0.0196098516782 ["Character/Dirichlet/837/506"] "1-837-837.508-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.508" [[1, 0.0]] [] 0 true true false false -0.02925824403648532 0 0.865854274409 ["Character/Dirichlet/837/508"] "1-837-837.511-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.511" [[1, 0.0]] [] 0 true true false false -0.35561198883590067 0 0.0223868676917 ["Character/Dirichlet/837/511"] "1-837-837.515-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.515" [[1, 0.0]] [] 0 true true false false -0.13411468644776858 0 0.478573149563 ["Character/Dirichlet/837/515"] "1-837-837.517-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.517" [[1, 0.0]] [] 0 true true false false -0.4902735691561651 0 0.0272902326646 ["Character/Dirichlet/837/517"] "1-837-837.52-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.52" [[1, 0.0]] [] 0 true true false false 0.15787457899198307 0 1.3598593558 ["Character/Dirichlet/837/52"] "1-837-837.520-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.520" [[1, 0.0]] [] 0 true true false false -0.06809069964502917 0 0.131721482637 ["Character/Dirichlet/837/520"] "1-837-837.524-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.524" [[1, 0.0]] [] 0 true true false false 0.17418771141770384 0 0.455415919414 ["Character/Dirichlet/837/524"] "1-837-837.526-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.526" [[1, 0.0]] [] 0 true true false false -0.49074074074074076 0 1.52473858606 ["Character/Dirichlet/837/526"] "1-837-837.536-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.536" [[1, 0.0]] [] 0 true true false false 0.2922390583516467 0 0.311205439078 ["Character/Dirichlet/837/536"] "1-837-837.538-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.538" [[1, 0.0]] [] 0 true true false false -0.19577269143137563 0 0.693028410677 ["Character/Dirichlet/837/538"] "1-837-837.544-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.544" [[1, 0.0]] [] 0 true true false false -0.4708939752352911 0 1.62848726549 ["Character/Dirichlet/837/544"] "1-837-837.545-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.545" [[1, 0.0]] [] 0 true true false false 0.4674480197811019 0 1.24635946822 ["Character/Dirichlet/837/545"] "1-837-837.553-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.553" [[1, 0.0]] [] 0 true true false false -0.3630587489543943 0 0.334664326458 ["Character/Dirichlet/837/553"] "1-837-837.556-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.556" [[1, 0.0]] [] 0 true true false false 0.003760136984048749 0 0.71256152327 ["Character/Dirichlet/837/556"] "1-837-837.56-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.56" [[1, 0.0]] [] 0 true true false false 0.30458173663391397 0 1.58359544752 ["Character/Dirichlet/837/56"] "1-837-837.560-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.560" [[1, 0.0]] [] 0 true true false false 0.40504890821848966 0 0.149288056718 ["Character/Dirichlet/837/560"] "1-837-837.563-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.563" [[1, 0.0]] [] 0 true true false false 0.028751596699419367 0 0.511459644311 ["Character/Dirichlet/837/563"] "1-837-837.571-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.571" [[1, 0.0]] [] 0 true true false false -0.28555657077832947 0 1.02091111321 ["Character/Dirichlet/837/571"] "1-837-837.572-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.572" [[1, 0.0]] [] 0 true true false false -0.06070114358982019 0 1.09477108816 ["Character/Dirichlet/837/572"] "1-837-837.578-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.578" [[1, 0.0]] [] 0 true true false false 0.06070114358982019 0 0.746630371107 ["Character/Dirichlet/837/578"] "1-837-837.58-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.58" [[1, 0.0]] [] 0 true true false false -0.19904313860621314 0 0.461467978144 ["Character/Dirichlet/837/58"] "1-837-837.580-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.580" [[1, 0.0]] [] 0 true true false false -0.2652426336883042 0 0.389870541036 ["Character/Dirichlet/837/580"] "1-837-837.59-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.59" [[1, 0.0]] [] 0 true true false false 0.359372896602889 0 1.44006491952 ["Character/Dirichlet/837/59"] "1-837-837.592-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.592" [[1, 0.0]] [] 0 true true false false -0.17639309751050156 0 0.461514966429 ["Character/Dirichlet/837/592"] "1-837-837.596-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.596" [[1, 0.0]] [] 0 true true false false 0.04109427498168663 0 0.805487332437 ["Character/Dirichlet/837/596"] "1-837-837.599-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.599" [[1, 0.0]] [] 0 true true false false -0.17418771141770384 0 0.167539393312 ["Character/Dirichlet/837/599"] "1-837-837.601-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.601" [[1, 0.0]] [] 0 true true false false 0.28555657077832947 0 1.41988431586 ["Character/Dirichlet/837/601"] "1-837-837.605-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.605" [[1, 0.0]] [] 0 true true false false 0.4097659065963252 0 1.26379959402 ["Character/Dirichlet/837/605"] "1-837-837.608-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.608" [[1, 0.0]] [] 0 true true false false 0.38440383207074996 0 1.02144481623 ["Character/Dirichlet/837/608"] "1-837-837.61-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.61" [[1, 0.0]] [] 0 true true false false 0.1574074074074074 0 1.05711315992 ["Character/Dirichlet/837/61"] "1-837-837.610-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.610" [[1, 0.0]] [] 0 true true false false -0.1754587543413503 0 0.123155708317 ["Character/Dirichlet/837/610"] "1-837-837.614-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.614" [[1, 0.0]] [] 0 true true false false -0.028751596699419367 0 0.504479887156 ["Character/Dirichlet/837/614"] "1-837-837.616-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.616" [[1, 0.0]] [] 0 true true false false -0.19904313860621314 0 0.00800466566832 ["Character/Dirichlet/837/616"] "1-837-837.617-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.617" [[1, 0.0]] [] 0 true true false false 0.359372896602889 0 0.919275354476 ["Character/Dirichlet/837/617"] "1-837-837.619-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.619" [[1, 0.0]] [] 0 true true false false 0.49074074074074076 0 0.488846801331 ["Character/Dirichlet/837/619"] "1-837-837.637-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.637" [[1, 0.0]] [] 0 true true false false -0.15607916042047623 0 0.865742419195 ["Character/Dirichlet/837/637"] "1-837-837.643-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.643" [[1, 0.0]] [] 0 true true false false -0.11577167620860168 0 0.806165034485 ["Character/Dirichlet/837/643"] "1-837-837.646-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.646" [[1, 0.0]] [] 0 true true false false -0.3815772674729128 0 0.046491155589 ["Character/Dirichlet/837/646"] "1-837-837.653-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.653" [[1, 0.0]] [] 0 true true false false -0.4097659065963252 0 0.157804171635 ["Character/Dirichlet/837/653"] "1-837-837.659-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.659" [[1, 0.0]] [] 0 true true false false 0.1634131770196839 0 0.555631948484 ["Character/Dirichlet/837/659"] "1-837-837.664-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.664" [[1, 0.0]] [] 0 true true false false 0.02925824403648532 0 0.834637711898 ["Character/Dirichlet/837/664"] "1-837-837.671-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.671" [[1, 0.0]] [] 0 true true false false 0.24588632877500538 0 1.26017001657 ["Character/Dirichlet/837/671"] "1-837-837.673-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.673" [[1, 0.0]] [] 0 true true false false -0.2837611522068227 0 1.71424550271 ["Character/Dirichlet/837/673"] "1-837-837.686-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.686" [[1, 0.0]] [] 0 true true false false -0.34859836220486906 0 1.57974135195 ["Character/Dirichlet/837/686"] "1-837-837.688-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.688" [[1, 0.0]] [] 0 true true false false 0.3815772674729128 0 0.958811400887 ["Character/Dirichlet/837/688"] "1-837-837.689-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.689" [[1, 0.0]] [] 0 true true false false -0.4403872064997949 0 0.97243502328 ["Character/Dirichlet/837/689"] "1-837-837.697-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.697" [[1, 0.0]] [] 0 true true false false 0.003760136984048749 0 0.987708269344 ["Character/Dirichlet/837/697"] "1-837-837.698-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.698" [[1, 0.0]] [] 0 true true false false -0.40504890821848966 0 1.37816942381 ["Character/Dirichlet/837/698"] "1-837-837.706-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.706" [[1, 0.0]] [] 0 true true false false -0.382905514459844 0 0.116209929917 ["Character/Dirichlet/837/706"] "1-837-837.707-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.707" [[1, 0.0]] [] 0 true true false false 0.1564335884857658 0 0.737864456553 ["Character/Dirichlet/837/707"] "1-837-837.709-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.709" [[1, 0.0]] [] 0 true true false false 0.11577167620860168 0 0.59364923789 ["Character/Dirichlet/837/709"] "1-837-837.722-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.722" [[1, 0.0]] [] 0 true true false false 0.10705387316646152 0 1.11985060508 ["Character/Dirichlet/837/722"] "1-837-837.724-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.724" [[1, 0.0]] [] 0 true true false false 0.15607916042047623 0 0.622124763197 ["Character/Dirichlet/837/724"] "1-837-837.731-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.731" [[1, 0.0]] [] 0 true true false false -0.38440383207074996 0 0.607826932859 ["Character/Dirichlet/837/731"] "1-837-837.736-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.736" [[1, 0.0]] [] 0 true true false false 0.19904313860621314 0 0.642634715186 ["Character/Dirichlet/837/736"] "1-837-837.742-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.742" [[1, 0.0]] [] 0 true true false false 0.35561198883590067 0 0.977809998509 ["Character/Dirichlet/837/742"] "1-837-837.749-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.749" [[1, 0.0]] [] 0 true true false false -0.1564335884857658 0 0.0707994764665 ["Character/Dirichlet/837/749"] "1-837-837.752-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.752" [[1, 0.0]] [] 0 true true false false 0.34859836220486906 0 1.15089355419 ["Character/Dirichlet/837/752"] "1-837-837.758-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.758" [[1, 0.0]] [] 0 true true false false 0.08744700455832798 0 0.755419191811 ["Character/Dirichlet/837/758"] "1-837-837.778-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.778" [[1, 0.0]] [] 0 true true false false 0.1754587543413503 0 0.617675965898 ["Character/Dirichlet/837/778"] "1-837-837.779-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.779" [[1, 0.0]] [] 0 true true false false -0.1634131770196839 0 0.60431257831 ["Character/Dirichlet/837/779"] "1-837-837.781-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.781" [[1, 0.0]] [] 0 true true false false 0.3630587489543943 0 0.977646968342 ["Character/Dirichlet/837/781"] "1-837-837.785-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.785" [[1, 0.0]] [] 0 true true false false -0.026039563269555675 0 0.351859828908 ["Character/Dirichlet/837/785"] "1-837-837.787-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.787" [[1, 0.0]] [] 0 true true false false -0.02925824403648532 0 0.332461239479 ["Character/Dirichlet/837/787"] "1-837-837.79-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.79" [[1, 0.0]] [] 0 true true false false 0.17725417291285708 0 0.392135911857 ["Character/Dirichlet/837/79"] "1-837-837.790-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.790" [[1, 0.0]] [] 0 true true false false 0.31105467783076607 0 0.136749029927 ["Character/Dirichlet/837/790"] "1-837-837.794-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.794" [[1, 0.0]] [] 0 true true false false -0.4674480197811019 0 0.0259732034322 ["Character/Dirichlet/837/794"] "1-837-837.796-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.796" [[1, 0.0]] [] 0 true true false false 0.17639309751050156 0 1.05855821513 ["Character/Dirichlet/837/796"] "1-837-837.799-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.799" [[1, 0.0]] [] 0 true true false false -0.06809069964502917 0 0.761937103238 ["Character/Dirichlet/837/799"] "1-837-837.803-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.803" [[1, 0.0]] [] 0 true true false false 0.17418771141770384 0 0.562287977402 ["Character/Dirichlet/837/803"] "1-837-837.805-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.805" [[1, 0.0]] [] 0 true true false false -0.1574074074074074 0 0.708012427454 ["Character/Dirichlet/837/805"] "1-837-837.815-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.815" [[1, 0.0]] [] 0 true true false false 0.2922390583516467 0 0.0888765105316 ["Character/Dirichlet/837/815"] "1-837-837.817-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.817" [[1, 0.0]] [] 0 true true false false 0.4708939752352911 0 0.0744868702903 ["Character/Dirichlet/837/817"] "1-837-837.823-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.823" [[1, 0.0]] [] 0 true true false false 0.19577269143137563 0 1.36308097821 ["Character/Dirichlet/837/823"] "1-837-837.824-r1-0-0" 89.94814205310114 89.94814205310114 1 837 "837.824" [[1, 0.0]] [] 0 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-- # 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).