# lfunc_search downloaded from the LMFDB on 01 May 2026. # Search link: https://www.lmfdb.org/L/1/287/287.188/r0-0 # Query "{'degree': 1, 'conductor': 287, 'spectral_label': 'r0-0'}" returned 98 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. "1-287-287.100-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.100" [[0, 0.0]] [] 0 true true false false -0.45755651376099693 0 2.61672558328 ["Character/Dirichlet/287/100"] "1-287-287.101-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.101" [[0, 0.0]] [] 0 true true false false -0.3152313867260041 0 0.111207306943 ["Character/Dirichlet/287/101"] "1-287-287.102-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.102" [[0, 0.0]] [] 0 true true false false 0.23195278983378084 0 1.26235739686 ["Character/Dirichlet/287/102"] "1-287-287.104-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.104" [[0, 0.0]] [] 0 true true false false -0.011689285719265497 0 1.15341411807 ["Character/Dirichlet/287/104"] "1-287-287.107-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.107" [[0, 0.0]] [] 0 true true false false -0.1525021546225351 0 0.804381686479 ["Character/Dirichlet/287/107"] "1-287-287.108-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.108" [[0, 0.0]] [] 0 true true false false 0.12697270197498212 0 0.995587477417 ["Character/Dirichlet/287/108"] "1-287-287.110-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.110" [[0, 0.0]] [] 0 true true false false -0.1751503340013473 0 1.53165474716 ["Character/Dirichlet/287/110"] "1-287-287.111-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.111" [[0, 0.0]] [] 0 true true false false 0.33802740459885516 0 1.85618065608 ["Character/Dirichlet/287/111"] "1-287-287.114-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.114" [[0, 0.0]] [] 0 true true false false 0.3730589650629591 0 2.34523774677 ["Character/Dirichlet/287/114"] "1-287-287.117-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.117" [[0, 0.0]] [] 0 true true false false -0.3022140031337662 0 0.557322389587 ["Character/Dirichlet/287/117"] "1-287-287.12-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.12" [[0, 0.0]] [] 0 true true false false 0.22706302656001306 0 1.77337339973 ["Character/Dirichlet/287/12"] "1-287-287.121-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.121" [[0, 0.0]] [] 0 true true false false -0.23195278983378084 0 0.850855882771 ["Character/Dirichlet/287/121"] "1-287-287.128-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.128" [[0, 0.0]] [] 0 true true false false -0.2594865646404779 0 1.10282466763 ["Character/Dirichlet/287/128"] "1-287-287.129-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.129" [[0, 0.0]] [] 0 true true false false 0.009198413151708941 0 1.30995574011 ["Character/Dirichlet/287/129"] "1-287-287.13-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.13" [[0, 0.0]] [] 0 true true false false 0.49792899355338316 0 2.51821904458 ["Character/Dirichlet/287/13"] "1-287-287.136-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.136" [[0, 0.0]] [] 0 true true false false -0.31108937383277036 0 0.830177839145 ["Character/Dirichlet/287/136"] "1-287-287.138-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.138" [[0, 0.0]] [] 0 true true false false -0.06826676894756155 0 1.21634915803 ["Character/Dirichlet/287/138"] "1-287-287.144-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.144" [[0, 0.0]] [] 0 true true false false -0.07482309231415873 0 1.19145031883 ["Character/Dirichlet/287/144"] "1-287-287.145-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.145" [[0, 0.0]] [] 0 true true false false -0.19852890543987828 0 0.823840150797 ["Character/Dirichlet/287/145"] "1-287-287.150-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.150" [[0, 0.0]] [] 0 true true false false 0.0038107370247308175 0 1.28150235794 ["Character/Dirichlet/287/150"] "1-287-287.152-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.152" [[0, 0.0]] [] 0 true true false false 0.15118778487824236 0 0.986563903143 ["Character/Dirichlet/287/152"] "1-287-287.153-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.153" [[0, 0.0]] [] 0 true true false false 0.05986691774563067 0 2.12258191553 ["Character/Dirichlet/287/153"] "1-287-287.156-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.156" [[0, 0.0]] [] 0 true true false false 0.11299285102775737 0 1.04672358506 ["Character/Dirichlet/287/156"] "1-287-287.157-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.157" [[0, 0.0]] [] 0 true true false false 0.3022140031337662 0 1.78024193699 ["Character/Dirichlet/287/157"] "1-287-287.16-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.16" [[0, 0.0]] [] 0 true true false false 0.31106280014827636 0 1.24370365305 ["Character/Dirichlet/287/16"] "1-287-287.163-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.163" [[0, 0.0]] [] 0 true true false false 0.4267531431936398 0 0.557383863456 ["Character/Dirichlet/287/163"] "1-287-287.167-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.167" [[0, 0.0]] [] 0 true true false false -0.13695617861466294 0 1.29189474459 ["Character/Dirichlet/287/167"] "1-287-287.17-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.17" [[0, 0.0]] [] 0 true true false false -0.15118778487824236 0 1.33776751988 ["Character/Dirichlet/287/17"] "1-287-287.171-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.171" [[0, 0.0]] [] 0 true true false false 0.3644808262895166 0 0.183052173466 ["Character/Dirichlet/287/171"] "1-287-287.172-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.172" [[0, 0.0]] [] 0 true true false false 0.06626244678841749 0 2.05431779178 ["Character/Dirichlet/287/172"] "1-287-287.178-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.178" [[0, 0.0]] [] 0 true true false false 0.3237957983352757 0 0.0259051127847 ["Character/Dirichlet/287/178"] "1-287-287.18-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.18" [[0, 0.0]] [] 0 true true false false -0.31106280014827636 0 0.914846669925 ["Character/Dirichlet/287/18"] "1-287-287.181-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.181" [[0, 0.0]] [] 0 true true false false -0.33802740459885516 0 0.601742371836 ["Character/Dirichlet/287/181"] "1-287-287.184-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.184" [[0, 0.0]] [] 0 true true false false 0.08545907622106032 0 1.76617959879 ["Character/Dirichlet/287/184"] "1-287-287.188-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.188" [[0, 0.0]] [] 0 true true false false -0.4139026462806259 0 0.0493791290223 ["Character/Dirichlet/287/188"] "1-287-287.19-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.19" [[0, 0.0]] [] 0 true true false false 0.31108937383277036 0 1.80097367572 ["Character/Dirichlet/287/19"] "1-287-287.192-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.192" [[0, 0.0]] [] 0 true true false false 0.19852890543987828 0 1.27360309259 ["Character/Dirichlet/287/192"] "1-287-287.194-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.194" [[0, 0.0]] [] 0 true true false false -0.12697270197498212 0 0.941620275433 ["Character/Dirichlet/287/194"] "1-287-287.199-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.199" [[0, 0.0]] [] 0 true true false false 0.07146523630745942 0 2.02869266522 ["Character/Dirichlet/287/199"] "1-287-287.2-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.2" [[0, 0.0]] [] 0 true true false false 0.07482309231415873 0 1.68994707955 ["Character/Dirichlet/287/2"] "1-287-287.200-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.200" [[0, 0.0]] [] 0 true true false false 0.08023126682430308 0 0.933238604421 ["Character/Dirichlet/287/200"] "1-287-287.202-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.202" [[0, 0.0]] [] 0 true true false false 0.1906503567453436 0 0.120555750906 ["Character/Dirichlet/287/202"] "1-287-287.207-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.207" [[0, 0.0]] [] 0 true true false false 0.22131680592687927 0 0.867389571735 ["Character/Dirichlet/287/207"] "1-287-287.208-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.208" [[0, 0.0]] [] 0 true true false false -0.3237957983352757 0 1.79253259357 ["Character/Dirichlet/287/208"] "1-287-287.214-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.214" [[0, 0.0]] [] 0 true true false false -0.3730589650629591 0 1.32022438192 ["Character/Dirichlet/287/214"] "1-287-287.216-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.216" [[0, 0.0]] [] 0 true true false false -0.11857285077305126 0 1.44403241009 ["Character/Dirichlet/287/216"] "1-287-287.220-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.220" [[0, 0.0]] [] 0 true true false false 0.30541247049366405 0 1.83293453396 ["Character/Dirichlet/287/220"] "1-287-287.221-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.221" [[0, 0.0]] [] 0 true true false false 0.45755651376099693 0 0.42040957126 ["Character/Dirichlet/287/221"] "1-287-287.222-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.222" [[0, 0.0]] [] 0 true true false false 0.47513297568053203 0 2.63882313763 ["Character/Dirichlet/287/222"] "1-287-287.226-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.226" [[0, 0.0]] [] 0 true true false false -0.22131680592687927 0 0.807602787219 ["Character/Dirichlet/287/226"] "1-287-287.227-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.227" [[0, 0.0]] [] 0 true true false false 0.1751503340013473 0 1.52889220513 ["Character/Dirichlet/287/227"] "1-287-287.228-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.228" [[0, 0.0]] [] 0 true true false false 0.1525021546225351 0 1.95238931579 ["Character/Dirichlet/287/228"] "1-287-287.229-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.229" [[0, 0.0]] [] 0 true true false false 0.39925773399876135 0 2.20635643592 ["Character/Dirichlet/287/229"] "1-287-287.23-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.23" [[0, 0.0]] [] 0 true true false false 0.0060084410098145725 0 1.45847765123 ["Character/Dirichlet/287/23"] "1-287-287.234-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.234" [[0, 0.0]] [] 0 true true false false -0.47513297568053203 0 0.958893961072 ["Character/Dirichlet/287/234"] "1-287-287.24-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.24" [[0, 0.0]] [] 0 true true false false -0.22706302656001306 0 1.29579085839 ["Character/Dirichlet/287/24"] "1-287-287.242-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.242" [[0, 0.0]] [] 0 true true false false 0.06819249766789849 0 1.3556967918 ["Character/Dirichlet/287/242"] "1-287-287.243-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.243" [[0, 0.0]] [] 0 true true false false 0.0038107370247308175 0 1.0377230295 ["Character/Dirichlet/287/243"] "1-287-287.25-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.25" [[0, 0.0]] [] 0 true true false false -0.0060084410098145725 0 1.52442752857 ["Character/Dirichlet/287/25"] "1-287-287.254-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.254" [[0, 0.0]] [] 0 true true false false -0.08023126682430308 0 0.573629835137 ["Character/Dirichlet/287/254"] "1-287-287.256-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.256" [[0, 0.0]] [] 0 true true false false 0.07830121594482206 0 1.37092480582 ["Character/Dirichlet/287/256"] "1-287-287.257-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.257" [[0, 0.0]] [] 0 true true false false -0.30541247049366405 0 0.972584716079 ["Character/Dirichlet/287/257"] "1-287-287.258-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.258" [[0, 0.0]] [] 0 true true false false 0.4139026462806259 0 1.66212235816 ["Character/Dirichlet/287/258"] "1-287-287.26-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.26" [[0, 0.0]] [] 0 true true false false -0.24670653746624346 0 0.581765595629 ["Character/Dirichlet/287/26"] "1-287-287.265-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.265" [[0, 0.0]] [] 0 true true false false 0.49792899355338316 0 0.658041612635 ["Character/Dirichlet/287/265"] "1-287-287.27-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.27" [[0, 0.0]] [] 0 true true false false -0.1906503567453436 0 2.62839574553 ["Character/Dirichlet/287/27"] "1-287-287.272-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.272" [[0, 0.0]] [] 0 true true false false -0.05986691774563067 0 1.68199789061 ["Character/Dirichlet/287/272"] "1-287-287.276-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.276" [[0, 0.0]] [] 0 true true false false 0.24670653746624346 0 1.25416516856 ["Character/Dirichlet/287/276"] "1-287-287.277-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.277" [[0, 0.0]] [] 0 true true false false -0.21611715928673814 0 0.601583718829 ["Character/Dirichlet/287/277"] "1-287-287.282-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.282" [[0, 0.0]] [] 0 true true false false -0.06626244678841749 0 1.64646403383 ["Character/Dirichlet/287/282"] "1-287-287.3-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.3" [[0, 0.0]] [] 0 true true false false 0.04988344110594987 0 1.73499082485 ["Character/Dirichlet/287/3"] "1-287-287.32-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.32" [[0, 0.0]] [] 0 true true false false -0.4804473213243204 0 0.498194332142 ["Character/Dirichlet/287/32"] "1-287-287.34-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.34" [[0, 0.0]] [] 0 true true false false 0.11537438341315336 0 1.18782186527 ["Character/Dirichlet/287/34"] "1-287-287.37-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.37" [[0, 0.0]] [] 0 true true false false -0.07830121594482206 0 0.471107245775 ["Character/Dirichlet/287/37"] "1-287-287.38-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.38" [[0, 0.0]] [] 0 true true false false 0.3774899764659564 0 2.38931316363 ["Character/Dirichlet/287/38"] "1-287-287.39-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.39" [[0, 0.0]] [] 0 true true false false -0.08545907622106032 0 0.785892974256 ["Character/Dirichlet/287/39"] "1-287-287.4-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.4" [[0, 0.0]] [] 0 true true false false 0.3626108728994587 0 2.28939254213 ["Character/Dirichlet/287/4"] "1-287-287.46-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.46" [[0, 0.0]] [] 0 true true false false -0.11299285102775737 0 1.17803252595 ["Character/Dirichlet/287/46"] "1-287-287.47-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.47" [[0, 0.0]] [] 0 true true false false -0.3644808262895166 0 2.01656407155 ["Character/Dirichlet/287/47"] "1-287-287.48-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.48" [[0, 0.0]] [] 0 true true false false 0.17764120656890386 0 1.35607854808 ["Character/Dirichlet/287/48"] "1-287-287.51-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.51" [[0, 0.0]] [] 0 true true false false -0.06819249766789849 0 1.33439679232 ["Character/Dirichlet/287/51"] "1-287-287.52-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.52" [[0, 0.0]] [] 0 true true false false 0.06826676894756155 0 0.514516571823 ["Character/Dirichlet/287/52"] "1-287-287.54-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.54" [[0, 0.0]] [] 0 true true false false 0.3152313867260041 0 1.84343173994 ["Character/Dirichlet/287/54"] "1-287-287.55-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.55" [[0, 0.0]] [] 0 true true false false 0.13695617861466294 0 1.28998706782 ["Character/Dirichlet/287/55"] "1-287-287.6-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.6" [[0, 0.0]] [] 0 true true false false -0.17764120656890386 0 1.42083186249 ["Character/Dirichlet/287/6"] "1-287-287.68-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.68" [[0, 0.0]] [] 0 true true false false -0.3774899764659564 0 1.11733482947 ["Character/Dirichlet/287/68"] "1-287-287.69-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.69" [[0, 0.0]] [] 0 true true false false 0.011689285719265497 0 0.834435247847 ["Character/Dirichlet/287/69"] "1-287-287.72-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.72" [[0, 0.0]] [] 0 true true false false -0.3626108728994587 0 1.14353016964 ["Character/Dirichlet/287/72"] "1-287-287.74-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.74" [[0, 0.0]] [] 0 true true false false 0.2594865646404779 0 1.8877395793 ["Character/Dirichlet/287/74"] "1-287-287.75-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.75" [[0, 0.0]] [] 0 true true false false -0.07146523630745942 0 1.53901451368 ["Character/Dirichlet/287/75"] "1-287-287.76-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.76" [[0, 0.0]] [] 0 true true false false -0.11537438341315336 0 0.850814066532 ["Character/Dirichlet/287/76"] "1-287-287.81-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.81" [[0, 0.0]] [] 0 true true false false -0.4267531431936398 0 2.1209899184 ["Character/Dirichlet/287/81"] "1-287-287.86-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.86" [[0, 0.0]] [] 0 true true false false 0.21611715928673814 0 1.71400594748 ["Character/Dirichlet/287/86"] "1-287-287.89-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.89" [[0, 0.0]] [] 0 true true false false -0.009198413151708941 0 1.12651864732 ["Character/Dirichlet/287/89"] "1-287-287.9-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.9" [[0, 0.0]] [] 0 true true false false 0.4804473213243204 0 2.01824382115 ["Character/Dirichlet/287/9"] "1-287-287.94-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.94" [[0, 0.0]] [] 0 true true false false -0.39925773399876135 0 0.499382776678 ["Character/Dirichlet/287/94"] "1-287-287.96-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.96" [[0, 0.0]] [] 0 true true false false -0.04988344110594987 0 1.60210282136 ["Character/Dirichlet/287/96"] "1-287-287.97-r0-0-0" 1.3328223966973418 1.3328223966973418 1 287 "287.97" [[0, 0.0]] [] 0 true true false false 0.11857285077305126 0 1.46610208172 ["Character/Dirichlet/287/97"] # 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).