# lfunc_search downloaded from the LMFDB on 30 May 2026. # Search link: https://www.lmfdb.org/L/1/261/261.67 # Query "{'degree': 1, 'conductor': 261}" returned 108 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. "1-261-261.101-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.101" [[0, 0.0]] [] 0 true true false false -0.009757254669130177 0 1.3886448119 ["Character/Dirichlet/261/101"] "1-261-261.103-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.103" [[0, 0.0]] [] 0 true true false false -0.04034897351653405 0 1.22186492136 ["Character/Dirichlet/261/103"] "1-261-261.104-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.104" [[0, 0.0]] [] 0 true true false false 0.1641647090467336 0 1.36685288139 ["Character/Dirichlet/261/104"] "1-261-261.11-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.11" [[0, 0.0]] [] 0 true true false false 0.37196594046340736 0 2.38886910974 ["Character/Dirichlet/261/11"] "1-261-261.112-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.112" [[0, 0.0]] [] 0 true true false false -0.05493795947197606 0 1.59471643123 ["Character/Dirichlet/261/112"] "1-261-261.113-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.113" [[0, 0.0]] [] 0 true true false false -0.03443444460037264 0 1.00817551133 ["Character/Dirichlet/261/113"] "1-261-261.115-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.115" [[0, 0.0]] [] 0 true true false false -0.4444444444444444 0 2.43548580834 ["Character/Dirichlet/261/115"] "1-261-261.119-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.119" [[0, 0.0]] [] 0 true true false false -0.2856902411657522 0 0.426994172374 ["Character/Dirichlet/261/119"] "1-261-261.121-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.121" [[0, 0.0]] [] 0 true true false false 0.10973077272287762 0 1.51619226899 ["Character/Dirichlet/261/121"] "1-261-261.128-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.128" [[0, 0.0]] [] 0 true true false false -0.1641647090467336 0 0.447734855479 ["Character/Dirichlet/261/128"] "1-261-261.13-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.13" [[0, 0.0]] [] 0 true true false false -0.09257372693228788 0 1.62333475879 ["Character/Dirichlet/261/13"] "1-261-261.131-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.131" [[0, 0.0]] [] 0 true true false false -0.29054601700059757 0 0.922129655628 ["Character/Dirichlet/261/131"] "1-261-261.137-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.137" [[0, 0.0]] [] 0 true true false false -0.007606102100740132 0 1.69409492273 ["Character/Dirichlet/261/137"] "1-261-261.139-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.139" [[0, 0.0]] [] 0 true true false false 0.18054174806786955 0 1.90687160531 ["Character/Dirichlet/261/139"] "1-261-261.14-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.14" [[0, 0.0]] [] 0 true true false false 0.37913163421975876 0 1.50883249437 ["Character/Dirichlet/261/14"] "1-261-261.151-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.151" [[0, 0.0]] [] 0 true true false false -0.10973077272287762 0 1.68787273573 ["Character/Dirichlet/261/151"] "1-261-261.155-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.155" [[0, 0.0]] [] 0 true true false false -0.10319864772313672 0 1.40936778075 ["Character/Dirichlet/261/155"] "1-261-261.16-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.16" [[0, 0.0]] [] 0 true true false false 0.07076213759457706 0 1.51074772767 ["Character/Dirichlet/261/16"] "1-261-261.164-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.164" [[0, 0.0]] [] 0 true true false false 0.03443444460037264 0 1.22653475629 ["Character/Dirichlet/261/164"] "1-261-261.169-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.169" [[0, 0.0]] [] 0 true true false false -0.18054174806786955 0 1.2197980232 ["Character/Dirichlet/261/169"] "1-261-261.176-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.176" [[0, 0.0]] [] 0 true true false false -0.3205650941105136 0 1.13095871181 ["Character/Dirichlet/261/176"] "1-261-261.178-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.178" [[0, 0.0]] [] 0 true true false false 0.42728739865385473 0 0.320334761666 ["Character/Dirichlet/261/178"] "1-261-261.182-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.182" [[0, 0.0]] [] 0 true true false false 0.2391451706477038 0 1.38018074413 ["Character/Dirichlet/261/182"] "1-261-261.185-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.185" [[0, 0.0]] [] 0 true true false false -0.2391451706477038 0 1.20837014936 ["Character/Dirichlet/261/185"] "1-261-261.187-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.187" [[0, 0.0]] [] 0 true true false false 0.018537384178823234 0 0.617840735916 ["Character/Dirichlet/261/187"] "1-261-261.191-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.191" [[0, 0.0]] [] 0 true true false false -0.22472417984215529 0 1.0394558071 ["Character/Dirichlet/261/191"] "1-261-261.196-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.196" [[0, 0.0]] [] 0 true true false false -0.31617628754274363 0 0.843186214766 ["Character/Dirichlet/261/196"] "1-261-261.2-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.2" [[0, 0.0]] [] 0 true true false false 0.29054601700059757 0 1.50228928598 ["Character/Dirichlet/261/2"] "1-261-261.200-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.200" [[0, 0.0]] [] 0 true true false false -0.4233233334892616 0 0.901271112636 ["Character/Dirichlet/261/200"] "1-261-261.202-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.202" [[0, 0.0]] [] 0 true true false false 0.4444444444444444 0 0.412198363145 ["Character/Dirichlet/261/202"] "1-261-261.218-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.218" [[0, 0.0]] [] 0 true true false false 0.3205650941105136 0 2.24436574448 ["Character/Dirichlet/261/218"] "1-261-261.22-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.22" [[0, 0.0]] [] 0 true true false false -0.42728739865385473 0 2.18366224079 ["Character/Dirichlet/261/22"] "1-261-261.221-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.221" [[0, 0.0]] [] 0 true true false false 0.007606102100740132 0 0.90755063307 ["Character/Dirichlet/261/221"] "1-261-261.223-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.223" [[0, 0.0]] [] 0 true true false false 0.04034897351653405 0 1.30878962453 ["Character/Dirichlet/261/223"] "1-261-261.230-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.230" [[0, 0.0]] [] 0 true true false false 0.009757254669130177 0 1.05416794486 ["Character/Dirichlet/261/230"] "1-261-261.238-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.238" [[0, 0.0]] [] 0 true true false false -0.22084188383398876 0 0.605844452199 ["Character/Dirichlet/261/238"] "1-261-261.241-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.241" [[0, 0.0]] [] 0 true true false false 0.09257372693228788 0 1.906938288 ["Character/Dirichlet/261/241"] "1-261-261.25-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.25" [[0, 0.0]] [] 0 true true false false 0.05617315163913506 0 0.951587355934 ["Character/Dirichlet/261/25"] "1-261-261.256-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.256" [[0, 0.0]] [] 0 true true false false -0.2916528591789807 0 0.835511896549 ["Character/Dirichlet/261/256"] "1-261-261.32-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.32" [[0, 0.0]] [] 0 true true false false 0.10319864772313672 0 1.98351076483 ["Character/Dirichlet/261/32"] "1-261-261.34-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.34" [[0, 0.0]] [] 0 true true false false 0.22084188383398876 0 1.37624301432 ["Character/Dirichlet/261/34"] "1-261-261.4-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.4" [[0, 0.0]] [] 0 true true false false 0.31617628754274363 0 1.66583529846 ["Character/Dirichlet/261/4"] "1-261-261.41-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.41" [[0, 0.0]] [] 0 true true false false 0.22472417984215529 0 1.83513808421 ["Character/Dirichlet/261/41"] "1-261-261.47-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.47" [[0, 0.0]] [] 0 true true false false -0.38128278678814875 0 0.211215408685 ["Character/Dirichlet/261/47"] "1-261-261.49-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.49" [[0, 0.0]] [] 0 true true false false -0.07076213759457706 0 0.533680277585 ["Character/Dirichlet/261/49"] "1-261-261.50-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.50" [[0, 0.0]] [] 0 true true false false 0.38128278678814875 0 1.40549197605 ["Character/Dirichlet/261/50"] "1-261-261.52-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.52" [[0, 0.0]] [] 0 true true false false 0.2916528591789807 0 1.65767460759 ["Character/Dirichlet/261/52"] "1-261-261.56-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.56" [[0, 0.0]] [] 0 true true false false -0.37913163421975876 0 0.650821543418 ["Character/Dirichlet/261/56"] "1-261-261.67-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.67" [[0, 0.0]] [] 0 true true false false -0.018537384178823234 0 1.11518979367 ["Character/Dirichlet/261/67"] "1-261-261.68-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.68" [[0, 0.0]] [] 0 true true false false 0.2856902411657522 0 2.15988098268 ["Character/Dirichlet/261/68"] "1-261-261.7-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.7" [[0, 0.0]] [] 0 true true false false 0.05493795947197606 0 1.39249443319 ["Character/Dirichlet/261/7"] "1-261-261.77-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.77" [[0, 0.0]] [] 0 true true false false 0.4233233334892616 0 2.32910118253 ["Character/Dirichlet/261/77"] "1-261-261.94-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.94" [[0, 0.0]] [] 0 true true false false -0.05617315163913506 0 1.43239063308 ["Character/Dirichlet/261/94"] "1-261-261.95-r0-0-0" 1.2120789043136104 1.2120789043136104 1 261 "261.95" [[0, 0.0]] [] 0 true true false false -0.37196594046340736 0 0.625897344352 ["Character/Dirichlet/261/95"] "1-261-261.106-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.106" [[1, 0.0]] [] 0 true true false false -0.21556555539962735 0 0.574428357878 ["Character/Dirichlet/261/106"] "1-261-261.110-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.110" [[1, 0.0]] [] 0 true true false false 0.04165285917898066 0 1.26087587302 ["Character/Dirichlet/261/110"] "1-261-261.122-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.122" [[1, 0.0]] [] 0 true true false false 0.029158116166011272 0 1.32915343828 ["Character/Dirichlet/261/122"] "1-261-261.124-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.124" [[1, 0.0]] [] 0 true true false false -0.010854829352296218 0 1.2082192352 ["Character/Dirichlet/261/124"] "1-261-261.130-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.130" [[1, 0.0]] [] 0 true true false false -0.3708683657802413 0 0.985531210162 ["Character/Dirichlet/261/130"] "1-261-261.133-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.133" [[1, 0.0]] [] 0 true true false false -0.4747241798421553 0 0.197881189611 ["Character/Dirichlet/261/133"] "1-261-261.140-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.140" [[1, 0.0]] [] 0 true true false false -0.04165285917898066 0 0.549257654158 ["Character/Dirichlet/261/140"] "1-261-261.142-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.142" [[1, 0.0]] [] 0 true true false false 0.3266766665107385 0 1.02196558384 ["Character/Dirichlet/261/142"] "1-261-261.148-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.148" [[1, 0.0]] [] 0 true true false false 0.3531986477231367 0 0.0331875234686 ["Character/Dirichlet/261/148"] "1-261-261.149-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.149" [[1, 0.0]] [] 0 true true false false -0.4338237124572564 0 0.596943062734 ["Character/Dirichlet/261/149"] "1-261-261.157-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.157" [[1, 0.0]] [] 0 true true false false 0.4747241798421553 0 1.80700481218 ["Character/Dirichlet/261/157"] "1-261-261.158-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.158" [[1, 0.0]] [] 0 true true false false 0.15742627306771215 0 1.04450545788 ["Character/Dirichlet/261/158"] "1-261-261.160-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.160" [[1, 0.0]] [] 0 true true false false 0.07056509411051358 0 0.850736179916 ["Character/Dirichlet/261/160"] "1-261-261.166-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.166" [[1, 0.0]] [] 0 true true false false -0.36871721321185125 0 0.583058236538 ["Character/Dirichlet/261/166"] "1-261-261.167-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.167" [[1, 0.0]] [] 0 true true false false -0.1772873986538547 0 0.667825079197 ["Character/Dirichlet/261/167"] "1-261-261.173-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.173" [[1, 0.0]] [] 0 true true false false 0.3055555555555556 0 0.157574525539 ["Character/Dirichlet/261/173"] "1-261-261.184-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.184" [[1, 0.0]] [] 0 true true false false -0.46430975883424785 0 1.77375403902 ["Character/Dirichlet/261/184"] "1-261-261.193-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.193" [[1, 0.0]] [] 0 true true false false -0.3266766665107385 0 0.480456348341 ["Character/Dirichlet/261/193"] "1-261-261.194-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.194" [[1, 0.0]] [] 0 true true false false 0.17923786240542294 0 1.39375132397 ["Character/Dirichlet/261/194"] "1-261-261.20-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.20" [[1, 0.0]] [] 0 true true false false -0.20965102648346598 0 0.747835773382 ["Character/Dirichlet/261/20"] "1-261-261.205-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.205" [[1, 0.0]] [] 0 true true false false 0.04054601700059754 0 1.25607900738 ["Character/Dirichlet/261/205"] "1-261-261.209-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.209" [[1, 0.0]] [] 0 true true false false -0.3597307727228776 0 1.52117498504 ["Character/Dirichlet/261/209"] "1-261-261.211-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.211" [[1, 0.0]] [] 0 true true false false -0.12196594046340734 0 0.619951560314 ["Character/Dirichlet/261/211"] "1-261-261.212-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.212" [[1, 0.0]] [] 0 true true false false 0.23146261582117678 0 1.06952331235 ["Character/Dirichlet/261/212"] "1-261-261.214-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.214" [[1, 0.0]] [] 0 true true false false 0.12196594046340734 0 1.15261314631 ["Character/Dirichlet/261/214"] "1-261-261.220-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.220" [[1, 0.0]] [] 0 true true false false 0.4141647090467336 0 2.09136406651 ["Character/Dirichlet/261/220"] "1-261-261.227-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.227" [[1, 0.0]] [] 0 true true false false -0.43054174806786955 0 2.05425479074 ["Character/Dirichlet/261/227"] "1-261-261.229-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.229" [[1, 0.0]] [] 0 true true false false 0.21556555539962735 0 1.43410969145 ["Character/Dirichlet/261/229"] "1-261-261.23-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.23" [[1, 0.0]] [] 0 true true false false 0.43054174806786955 0 0.142216073638 ["Character/Dirichlet/261/23"] "1-261-261.236-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.236" [[1, 0.0]] [] 0 true true false false 0.1772873986538547 0 0.577281634116 ["Character/Dirichlet/261/236"] "1-261-261.239-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.239" [[1, 0.0]] [] 0 true true false false 0.19382684836086495 0 0.527260633096 ["Character/Dirichlet/261/239"] "1-261-261.245-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.245" [[1, 0.0]] [] 0 true true false false -0.23146261582117678 0 0.889895038846 ["Character/Dirichlet/261/245"] "1-261-261.247-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.247" [[1, 0.0]] [] 0 true true false false -0.04054601700059754 0 0.702026871043 ["Character/Dirichlet/261/247"] "1-261-261.248-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.248" [[1, 0.0]] [] 0 true true false false 0.20965102648346598 0 1.79566814213 ["Character/Dirichlet/261/248"] "1-261-261.250-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.250" [[1, 0.0]] [] 0 true true false false 0.36871721321185125 0 2.05102324926 ["Character/Dirichlet/261/250"] "1-261-261.254-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.254" [[1, 0.0]] [] 0 true true false false 0.4338237124572564 0 1.89083069729 ["Character/Dirichlet/261/254"] "1-261-261.257-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.257" [[1, 0.0]] [] 0 true true false false 0.19506204052802398 0 1.09251116967 ["Character/Dirichlet/261/257"] "1-261-261.259-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.259" [[1, 0.0]] [] 0 true true false false 0.3708683657802413 0 1.48714564072 ["Character/Dirichlet/261/259"] "1-261-261.31-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.31" [[1, 0.0]] [] 0 true true false false -0.07056509411051358 0 0.380578953303 ["Character/Dirichlet/261/31"] "1-261-261.38-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.38" [[1, 0.0]] [] 0 true true false false -0.15742627306771215 0 0.836595399303 ["Character/Dirichlet/261/38"] "1-261-261.40-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.40" [[1, 0.0]] [] 0 true true false false 0.010854829352296218 0 0.441629328201 ["Character/Dirichlet/261/40"] "1-261-261.43-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.43" [[1, 0.0]] [] 0 true true false false -0.2597572546691302 0 2.39537862894 ["Character/Dirichlet/261/43"] "1-261-261.5-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.5" [[1, 0.0]] [] 0 true true false false 0.3597307727228776 0 0.0444634588038 ["Character/Dirichlet/261/5"] "1-261-261.61-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.61" [[1, 0.0]] [] 0 true true false false 0.46430975883424785 0 0.275530489253 ["Character/Dirichlet/261/61"] "1-261-261.65-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.65" [[1, 0.0]] [] 0 true true false false -0.19506204052802398 0 0.620153531112 ["Character/Dirichlet/261/65"] "1-261-261.70-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.70" [[1, 0.0]] [] 0 true true false false -0.4141647090467336 0 0.768016740069 ["Character/Dirichlet/261/70"] "1-261-261.74-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.74" [[1, 0.0]] [] 0 true true false false -0.17923786240542294 0 0.297705481738 ["Character/Dirichlet/261/74"] "1-261-261.76-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.76" [[1, 0.0]] [] 0 true true false false 0.25760610210074014 0 1.55876461491 ["Character/Dirichlet/261/76"] "1-261-261.79-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.79" [[1, 0.0]] [] 0 true true false false -0.25760610210074014 0 0.298448638523 ["Character/Dirichlet/261/79"] "1-261-261.83-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.83" [[1, 0.0]] [] 0 true true false false -0.19382684836086495 0 0.4182164727 ["Character/Dirichlet/261/83"] "1-261-261.85-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.85" [[1, 0.0]] [] 0 true true false false 0.2597572546691302 0 0.0141180514468 ["Character/Dirichlet/261/85"] "1-261-261.86-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.86" [[1, 0.0]] [] 0 true true false false -0.3055555555555556 0 1.8840546159 ["Character/Dirichlet/261/86"] "1-261-261.92-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.92" [[1, 0.0]] [] 0 true true false false -0.029158116166011272 0 1.39550931088 ["Character/Dirichlet/261/92"] "1-261-261.97-r1-0-0" 28.04834537139713 28.04834537139713 1 261 "261.97" [[1, 0.0]] [] 0 true true false false -0.3531986477231367 0 1.68727051062 ["Character/Dirichlet/261/97"] # 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).