# lfunc_search downloaded from the LMFDB on 26 July 2026. # Search link: https://www.lmfdb.org/L/1/143/143.57 # Query "{'degree': 1, 'conductor': 143}" returned 99 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. "1-143-143.103-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.103" [[0, 0.0]] [] 0 true true false false -0.19621626289782826 0 1.0707314353 ["Character/Dirichlet/143/103"] "1-143-143.106-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.106" [[0, 0.0]] [] 0 true true false false -0.4151712251223647 0 0.999152840173 ["Character/Dirichlet/143/106"] "1-143-143.108-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.108" [[0, 0.0]] [] 0 true true false false -0.2906761786770401 0 1.03556202668 ["Character/Dirichlet/143/108"] "1-143-143.109-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.109" [[0, 0.0]] [] 0 true true false false 0.046791760452749714 0 1.31062539012 ["Character/Dirichlet/143/109"] "1-143-143.112-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.112" [[0, 0.0]] [] 0 true true false false -0.4002908932809712 0 0.306414066784 ["Character/Dirichlet/143/112"] "1-143-143.113-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.113" [[0, 0.0]] [] 0 true true false false 0.15352912053487827 0 1.87326742433 ["Character/Dirichlet/143/113"] "1-143-143.114-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.114" [[0, 0.0]] [] 0 true true false false 0.051742785709605034 0 2.27926305338 ["Character/Dirichlet/143/114"] "1-143-143.123-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.123" [[0, 0.0]] [] 0 true true false false 0.3070219643560788 0 1.77392717606 ["Character/Dirichlet/143/123"] "1-143-143.126-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.126" [[0, 0.0]] [] 0 true true false false 2.9987706656768033e-05 0 1.8661122081 ["Character/Dirichlet/143/126"] "1-143-143.128-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.128" [[0, 0.0]] [] 0 true true false false 0.17841229578313478 0 1.82279149631 ["Character/Dirichlet/143/128"] "1-143-143.138-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.138" [[0, 0.0]] [] 0 true true false false 0.3067073723754718 0 2.37413480561 ["Character/Dirichlet/143/138"] "1-143-143.16-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.16" [[0, 0.0]] [] 0 true true false false 0.0854042724325568 0 1.33746408528 ["Character/Dirichlet/143/16"] "1-143-143.18-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.18" [[0, 0.0]] [] 0 true true false false -0.11427484657981525 0 0.782901858007 ["Character/Dirichlet/143/18"] "1-143-143.19-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.19" [[0, 0.0]] [] 0 true true false false -0.17841229578313478 0 1.4874041337 ["Character/Dirichlet/143/19"] "1-143-143.2-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.2" [[0, 0.0]] [] 0 true true false false 0.39239624908197884 0 2.81394396712 ["Character/Dirichlet/143/2"] "1-143-143.21-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.21" [[0, 0.0]] [] 0 true true false false -0.046791760452749714 0 1.99186038023 ["Character/Dirichlet/143/21"] "1-143-143.24-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.24" [[0, 0.0]] [] 0 true true false false -0.014020230012521772 0 1.60029256159 ["Character/Dirichlet/143/24"] "1-143-143.25-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.25" [[0, 0.0]] [] 0 true true false false 0.19621626289782826 0 1.13377079502 ["Character/Dirichlet/143/25"] "1-143-143.28-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.28" [[0, 0.0]] [] 0 true true false false 0.09939451473842179 0 1.92486661247 ["Character/Dirichlet/143/28"] "1-143-143.3-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.3" [[0, 0.0]] [] 0 true true false false 0.2389034052607783 0 2.29506060886 ["Character/Dirichlet/143/3"] "1-143-143.32-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.32" [[0, 0.0]] [] 0 true true false false -0.33947890281569976 0 1.27027810888 ["Character/Dirichlet/143/32"] "1-143-143.36-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.36" [[0, 0.0]] [] 0 true true false false 0.4441753115052616 0 0.620826703529 ["Character/Dirichlet/143/36"] "1-143-143.38-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.38" [[0, 0.0]] [] 0 true true false false 0.04271713006960679 0 2.35893887997 ["Character/Dirichlet/143/38"] "1-143-143.4-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.4" [[0, 0.0]] [] 0 true true false false -0.4441753115052616 0 2.43794463942 ["Character/Dirichlet/143/4"] "1-143-143.41-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.41" [[0, 0.0]] [] 0 true true false false -0.09303801105723475 0 1.23306566586 ["Character/Dirichlet/143/41"] "1-143-143.42-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.42" [[0, 0.0]] [] 0 true true false false 2.9987706656768033e-05 0 2.0137371422 ["Character/Dirichlet/143/42"] "1-143-143.46-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.46" [[0, 0.0]] [] 0 true true false false -0.09939451473842179 0 1.16827023557 ["Character/Dirichlet/143/46"] "1-143-143.48-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.48" [[0, 0.0]] [] 0 true true false false -0.2389034052607783 0 1.12959939952 ["Character/Dirichlet/143/48"] "1-143-143.49-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.49" [[0, 0.0]] [] 0 true true false false 0.2906761786770401 0 1.65484880343 ["Character/Dirichlet/143/49"] "1-143-143.50-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.50" [[0, 0.0]] [] 0 true true false false -0.3070219643560788 0 0.62819300418 ["Character/Dirichlet/143/50"] "1-143-143.54-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.54" [[0, 0.0]] [] 0 true true false false -0.24589538191020033 0 0.524098281671 ["Character/Dirichlet/143/54"] "1-143-143.57-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.57" [[0, 0.0]] [] 0 true true false false -0.3067073723754718 0 1.52643329984 ["Character/Dirichlet/143/57"] "1-143-143.6-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.6" [[0, 0.0]] [] 0 true true false false 0.014020230012521772 0 1.18571385808 ["Character/Dirichlet/143/6"] "1-143-143.63-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.63" [[0, 0.0]] [] 0 true true false false 0.4994544901517354 0 0.668002155372 ["Character/Dirichlet/143/63"] "1-143-143.64-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.64" [[0, 0.0]] [] 0 true true false false -0.04271713006960679 0 1.61470811452 ["Character/Dirichlet/143/64"] "1-143-143.69-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.69" [[0, 0.0]] [] 0 true true false false -0.051742785709605034 0 1.93599533071 ["Character/Dirichlet/143/69"] "1-143-143.7-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.7" [[0, 0.0]] [] 0 true true false false 0.09303801105723475 0 1.68560080492 ["Character/Dirichlet/143/7"] "1-143-143.72-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.72" [[0, 0.0]] [] 0 true true false false -0.39239624908197884 0 1.24376827077 ["Character/Dirichlet/143/72"] "1-143-143.73-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.73" [[0, 0.0]] [] 0 true true false false -0.20785836748531467 0 1.36577911433 ["Character/Dirichlet/143/73"] "1-143-143.75-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.75" [[0, 0.0]] [] 0 true true false false 0.20524191853782656 0 1.77769552624 ["Character/Dirichlet/143/75"] "1-143-143.76-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.76" [[0, 0.0]] [] 0 true true false false 0.33947890281569976 0 2.62402277341 ["Character/Dirichlet/143/76"] "1-143-143.8-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.8" [[0, 0.0]] [] 0 true true false false 0.11427484657981525 0 1.15252202753 ["Character/Dirichlet/143/8"] "1-143-143.81-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.81" [[0, 0.0]] [] 0 true true false false -0.15352912053487827 0 1.44064171507 ["Character/Dirichlet/143/81"] "1-143-143.82-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.82" [[0, 0.0]] [] 0 true true false false -0.20524191853782656 0 1.19913635269 ["Character/Dirichlet/143/82"] "1-143-143.83-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.83" [[0, 0.0]] [] 0 true true false false 0.4002908932809712 0 2.53869593337 ["Character/Dirichlet/143/83"] "1-143-143.84-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.84" [[0, 0.0]] [] 0 true true false false 0.4994544901517354 0 3.2459841503 ["Character/Dirichlet/143/84"] "1-143-143.85-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.85" [[0, 0.0]] [] 0 true true false false 0.4151712251223647 0 2.62415584757 ["Character/Dirichlet/143/85"] "1-143-143.9-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.9" [[0, 0.0]] [] 0 true true false false -0.0854042724325568 0 0.30916625773 ["Character/Dirichlet/143/9"] "1-143-143.96-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.96" [[0, 0.0]] [] 0 true true false false 0.20785836748531467 0 2.16692667209 ["Character/Dirichlet/143/96"] "1-143-143.98-r0-0-0" 0.6640892081105222 0.6640892081105222 1 143 "143.98" [[0, 0.0]] [] 0 true true false false 0.24589538191020033 0 1.51752741337 ["Character/Dirichlet/143/98"] "1-143-143.10-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.10" [[1, 0.0]] [] 0 true true false false 0.2479590486074333 0 0.649255394048 ["Character/Dirichlet/143/10"] "1-143-143.101-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.101" [[1, 0.0]] [] 0 true true false false -0.10554008422078819 0 1.46805262447 ["Character/Dirichlet/143/101"] "1-143-143.102-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.102" [[1, 0.0]] [] 0 true true false false 0.035695165713527995 0 0.851555815712 ["Character/Dirichlet/143/102"] "1-143-143.107-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.107" [[1, 0.0]] [] 0 true true false false 0.20375374939551497 0 1.48720968992 ["Character/Dirichlet/143/107"] "1-143-143.115-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.115" [[1, 0.0]] [] 0 true true false false -0.1178039671146935 0 1.14019727113 ["Character/Dirichlet/143/115"] "1-143-143.116-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.116" [[1, 0.0]] [] 0 true true false false 0.3534991328282215 0 1.48799916374 ["Character/Dirichlet/143/116"] "1-143-143.119-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.119" [[1, 0.0]] [] 0 true true false false 0.20317825184059354 0 1.93999384161 ["Character/Dirichlet/143/119"] "1-143-143.120-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.120" [[1, 0.0]] [] 0 true true false false 0.042687142362950015 0 1.43285909445 ["Character/Dirichlet/143/120"] "1-143-143.124-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.124" [[1, 0.0]] [] 0 true true false false -0.04967911901237204 0 1.10294426981 ["Character/Dirichlet/143/124"] "1-143-143.125-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.125" [[1, 0.0]] [] 0 true true false false 0.256991976649422 0 1.42435370016 ["Character/Dirichlet/143/125"] "1-143-143.127-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.127" [[1, 0.0]] [] 0 true true false false -0.39854181856434523 0 0.402967855454 ["Character/Dirichlet/143/127"] "1-143-143.129-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.129" [[1, 0.0]] [] 0 true true false false 0.16106660703256495 0 1.08974519971 ["Character/Dirichlet/143/129"] "1-143-143.134-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.134" [[1, 0.0]] [] 0 true true false false 0.39854181856434523 0 1.14947866161 ["Character/Dirichlet/143/134"] "1-143-143.135-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.135" [[1, 0.0]] [] 0 true true false false -0.256991976649422 0 0.525248166868 ["Character/Dirichlet/143/135"] "1-143-143.136-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.136" [[1, 0.0]] [] 0 true true false false -0.035695165713527995 0 0.729221640424 ["Character/Dirichlet/143/136"] "1-143-143.137-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.137" [[1, 0.0]] [] 0 true true false false -0.20317825184059354 0 0.77677006441 ["Character/Dirichlet/143/137"] "1-143-143.139-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.139" [[1, 0.0]] [] 0 true true false false -0.20375374939551497 0 0.14034058436 ["Character/Dirichlet/143/139"] "1-143-143.140-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.140" [[1, 0.0]] [] 0 true true false false 0.40902565563999826 0 1.84711440417 ["Character/Dirichlet/143/140"] "1-143-143.141-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.141" [[1, 0.0]] [] 0 true true false false -0.20323822725390706 0 0.123648517252 ["Character/Dirichlet/143/141"] "1-143-143.142-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.142" [[1, 0.0]] [] 0 true true true true 0.0 0 1.61518022087 ["Character/Dirichlet/143/142"] "1-143-143.15-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.15" [[1, 0.0]] [] 0 true true false false 0.04967911901237204 0 0.834668406403 ["Character/Dirichlet/143/15"] "1-143-143.17-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.17" [[1, 0.0]] [] 0 true true false false 0.10554008422078819 0 2.03401016576 ["Character/Dirichlet/143/17"] "1-143-143.20-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.20" [[1, 0.0]] [] 0 true true false false -0.2886125119798071 0 0.233198098039 ["Character/Dirichlet/143/20"] "1-143-143.29-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.29" [[1, 0.0]] [] 0 true true false false -0.11837946466961494 0 1.29412446731 ["Character/Dirichlet/143/29"] "1-143-143.30-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.30" [[1, 0.0]] [] 0 true true false false -0.08689244157486838 0 0.902054182663 ["Character/Dirichlet/143/30"] "1-143-143.31-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.31" [[1, 0.0]] [] 0 true true false false 0.4959253696168571 0 1.75898214192 ["Character/Dirichlet/143/31"] "1-143-143.35-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.35" [[1, 0.0]] [] 0 true true false false -0.3961862751911715 0 0.327843071166 ["Character/Dirichlet/143/35"] "1-143-143.37-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.37" [[1, 0.0]] [] 0 true true false false -0.3567373600821286 0 1.07996440502 ["Character/Dirichlet/143/37"] "1-143-143.43-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.43" [[1, 0.0]] [] 0 true true false false -0.2479590486074333 0 0.534879966595 ["Character/Dirichlet/143/43"] "1-143-143.47-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.47" [[1, 0.0]] [] 0 true true false false -0.35057549755492146 0 2.76125168562 ["Character/Dirichlet/143/47"] "1-143-143.5-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.5" [[1, 0.0]] [] 0 true true false false -0.4104911094776435 0 2.73991565024 ["Character/Dirichlet/143/5"] "1-143-143.51-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.51" [[1, 0.0]] [] 0 true true false false -0.16106660703256495 0 0.0692664479369 ["Character/Dirichlet/143/51"] "1-143-143.58-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.58" [[1, 0.0]] [] 0 true true false false 0.3567373600821286 0 1.89782888307 ["Character/Dirichlet/143/58"] "1-143-143.59-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.59" [[1, 0.0]] [] 0 true true false false -0.4421116448080286 0 2.68929916144 ["Character/Dirichlet/143/59"] "1-143-143.60-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.60" [[1, 0.0]] [] 0 true true false false 0.4959253696168571 0 0.0859188833704 ["Character/Dirichlet/143/60"] "1-143-143.61-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.61" [[1, 0.0]] [] 0 true true false false 0.31081199046527147 0 0.144740866737 ["Character/Dirichlet/143/61"] "1-143-143.62-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.62" [[1, 0.0]] [] 0 true true false false 0.08689244157486838 0 1.41877176226 ["Character/Dirichlet/143/62"] "1-143-143.68-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.68" [[1, 0.0]] [] 0 true true false false -0.31081199046527147 0 2.3590901804 ["Character/Dirichlet/143/68"] "1-143-143.70-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.70" [[1, 0.0]] [] 0 true true false false 0.35057549755492146 0 0.132298823253 ["Character/Dirichlet/143/70"] "1-143-143.71-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.71" [[1, 0.0]] [] 0 true true false false 0.20323822725390706 0 1.8686299962 ["Character/Dirichlet/143/71"] "1-143-143.74-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.74" [[1, 0.0]] [] 0 true true false false 0.11837946466961494 0 2.09607761647 ["Character/Dirichlet/143/74"] "1-143-143.80-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.80" [[1, 0.0]] [] 0 true true false false 0.4421116448080286 0 0.605746594561 ["Character/Dirichlet/143/80"] "1-143-143.86-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.86" [[1, 0.0]] [] 0 true true false false 0.4104911094776435 0 0.71289791187 ["Character/Dirichlet/143/86"] "1-143-143.87-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.87" [[1, 0.0]] [] 0 true true false false -0.042687142362950015 0 0.858850652382 ["Character/Dirichlet/143/87"] "1-143-143.90-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.90" [[1, 0.0]] [] 0 true true false false -0.3534991328282215 0 0.796228802371 ["Character/Dirichlet/143/90"] "1-143-143.93-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.93" [[1, 0.0]] [] 0 true true false false 0.2886125119798071 0 0.984080478799 ["Character/Dirichlet/143/93"] "1-143-143.94-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.94" [[1, 0.0]] [] 0 true true false false 0.3961862751911715 0 1.13057446023 ["Character/Dirichlet/143/94"] "1-143-143.95-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.95" [[1, 0.0]] [] 0 true true false false -0.40902565563999826 0 0.283177155106 ["Character/Dirichlet/143/95"] "1-143-143.97-r1-0-0" 15.367484245631378 15.367484245631378 1 143 "143.97" [[1, 0.0]] [] 0 true true false false 0.1178039671146935 0 1.43515472527 ["Character/Dirichlet/143/97"] # 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).