# lfunc_search downloaded from the LMFDB on 02 May 2026. # Search link: https://www.lmfdb.org/L/2/7^2/49.8/c3-0 # Query "{'degree': 2, 'conductor': 49, 'spectral_label': 'c3-0'}" returned 258 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. "2-7e2-1.1-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 0.899811362693603773500375190422 ["ModularForm/GL2/Q/holomorphic/49/4/a/e/1/1"] "2-7e2-1.1-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 1.75457033817401880998895302335 ["ModularForm/GL2/Q/holomorphic/49/4/a/e/1/2"] "2-7e2-1.1-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 2.50258280312093899836671041842 ["ModularForm/GL2/Q/holomorphic/49/4/a/a/1/1", "ModularForm/GL2/Q/holomorphic/49/4/a/a"] "2-7e2-1.1-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.0 0 2.61045824242721581282890795060 ["ModularForm/GL2/Q/holomorphic/49/4/a/d/1/1", "ModularForm/GL2/Q/holomorphic/49/4/a/d"] "2-7e2-1.1-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 2.61622532863461852539181307197 ["ModularForm/GL2/Q/holomorphic/49/4/a/e/1/3"] "2-7e2-1.1-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true false true 0.0 0 3.07853439106705805267098237945 ["ModularForm/GL2/Q/holomorphic/49/4/a/e/1/4"] "2-7e2-1.1-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 3.47125000870806456222294388325 ["ModularForm/GL2/Q/holomorphic/49/4/a/b/1/1", "ModularForm/GL2/Q/holomorphic/49/4/a/b"] "2-7e2-1.1-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "1.1" [] [[1.5, 0.0]] 3 true true true true 0.5 1 3.94043229564655617480298696766 ["ModularForm/GL2/Q/holomorphic/49/4/a/c/1/1", "ModularForm/GL2/Q/holomorphic/49/4/a/c"] "2-7e2-49.11-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.43250783024241407 0 0.19064241085713251430297971178 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/4"] "2-7e2-49.11-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.30207999955485465 0 0.52476982214939754760309454839 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/10"] "2-7e2-49.11-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.158903046518286 0 3.20641662457481825666072576033 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/11"] "2-7e2-49.11-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.3406917814900678 0 3.55509103682028065490573077295 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/12"] "2-7e2-49.11-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.2610477186910829 0 3.68285594581546383849746395111 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/13"] "2-7e2-49.11-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.42621583685961456 0 0.77617105870005371960383911273 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/3"] "2-7e2-49.11-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.25128225797713205 0 0.911371202876487838968654359305 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/8"] "2-7e2-49.11-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.05618599463746324 0 1.16253748213492785395638128522 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/1"] "2-7e2-49.11-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.12058895130213032 0 1.74442649845016798806011608843 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/7"] "2-7e2-49.11-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.18520910766600937 0 1.88858917822755611147460754031 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/5"] "2-7e2-49.11-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.014890836389362195 0 1.98696775186407971791901725281 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/2"] "2-7e2-49.11-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false -0.04443736777186531 0 2.26917714157324523525398823412 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/6"] "2-7e2-49.11-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.11" [] [[1.5, 0.0]] 3 true true false false 0.11469497192650975 0 2.62852258414216789997756802388 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/11/9"] "2-7e2-49.15-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.33206554145966405 0 0.44826609500619063134153523109 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/1"] "2-7e2-49.15-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.40219565328608964 0 0.75789246153369928942320112311 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/4"] "2-7e2-49.15-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.33884220728907977 0 3.33521870413544123588829781808 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/11"] "2-7e2-49.15-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.4077737459659296 0 4.04024900443574914668189952666 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/12"] "2-7e2-49.15-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.3467433483961848 0 4.70082273974654956884269764021 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/13"] "2-7e2-49.15-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.386136407826744 0 0.909220272610984490028241901184 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/6"] "2-7e2-49.15-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.16966664213348265 0 1.04843766827590131206218842948 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/9"] "2-7e2-49.15-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.3640755738948678 0 1.10804864689591873558878203515 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/3"] "2-7e2-49.15-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.13258276786824502 0 1.82233001283256547054512356601 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/5"] "2-7e2-49.15-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.13313621167384276 0 2.16481855343906270035138184990 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/10"] "2-7e2-49.15-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.05869593208000966 0 2.28167039176107155911456164282 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/7"] "2-7e2-49.15-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false -0.04338310375973771 0 2.63180069809304397752291634513 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/2"] "2-7e2-49.15-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.15" [] [[1.5, 0.0]] 3 true true false false 0.1678377357981158 0 2.94964959403723049411770134239 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/15/8"] "2-7e2-49.16-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.25784825967608305 0 0.42004397336755907633541027702 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/3"] "2-7e2-49.16-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.39571567068004976 0 1.28672183957011908863136211025 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/12"] "2-7e2-49.16-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.32974071444741865 0 3.50607939578151755360257766926 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/7"] "2-7e2-49.16-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.44863325632254203 0 3.61685096352754956989021428486 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/4"] "2-7e2-49.16-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.3246339017673209 0 4.65643374910172233294327930075 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/1"] "2-7e2-49.16-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.3741856748213888 0 1.32661803939203161952727785875 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/10"] "2-7e2-49.16-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.09477264825457603 0 1.50461306381194860846776147945 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/2"] "2-7e2-49.16-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.2613070693333806 0 1.60315165707218578277334071845 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/8"] "2-7e2-49.16-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false -0.4084479479539141 0 1.93482494130002636738518603323 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/13"] "2-7e2-49.16-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.04414539990040816 0 2.09955251783646278224347258404 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/6"] "2-7e2-49.16-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.013378336685975663 0 2.45962147318069270227788345559 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/9"] "2-7e2-49.16-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.00392140179599096 0 2.55570535011217050179263329879 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/11"] "2-7e2-49.16-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.16" [] [[1.5, 0.0]] 3 true true false false 0.30324374237910007 0 2.67835352325746695203714714309 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/16/5"] "2-7e2-49.2-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.25238314546039275 0 0.50085459604258001036699105867 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/4"] "2-7e2-49.2-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.4239901477640005 0 0.55959435327544390859091381613 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/9"] "2-7e2-49.2-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.019782553529874066 0 3.42401508994909413209800488976 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/13"] "2-7e2-49.2-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.449168920702525 0 3.57399589722979073496871454656 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/7"] "2-7e2-49.2-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.45065841368525894 0 3.95209612526258231483164031451 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/5"] "2-7e2-49.2-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.02422790709107947 0 0.947528009585412750715983779929 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/1"] "2-7e2-49.2-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.0019666651265298775 0 1.58568423763844416524407897177 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/6"] "2-7e2-49.2-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.3248994699247852 0 1.78928952961235475965126534547 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/3"] "2-7e2-49.2-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.21411371686077346 0 1.87367624327371701053199140624 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/12"] "2-7e2-49.2-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false -0.11218926106126008 0 2.30488080101264535237708938452 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/11"] "2-7e2-49.2-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.3463460993083687 0 2.42427216194204550300375871924 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/2"] "2-7e2-49.2-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.07335682674001091 0 3.04032958566489573899557636588 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/10"] "2-7e2-49.2-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.2" [] [[1.5, 0.0]] 3 true true false false 0.2185600978691488 0 3.12213141834360180372773701355 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/2/8"] "2-7e2-49.22-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false 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2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false 0.38772608046355655 0 0.75435440811180382054942503232 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/22/3"] "2-7e2-49.22-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false -0.2537050772096601 0 0.911125756371785001192170224024 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/22/6"] "2-7e2-49.22-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false -0.2408812383023767 0 1.11192840962643918710918086905 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/22/8"] "2-7e2-49.22-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false -0.2202659646847544 0 1.27204763371891967383356458978 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/22/9"] "2-7e2-49.22-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.22" [] [[1.5, 0.0]] 3 true true false false -0.09674545811800407 0 1.35217221025532789944767132618 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"49.44" [] [[1.5, 0.0]] 3 true true false false 0.18346064250186572 0 2.25508182561442221590529429226 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/44/5"] "2-7e2-49.44-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.44" [] [[1.5, 0.0]] 3 true true false false -0.04870210649662606 0 2.62801276313358161725661793894 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/44/10"] "2-7e2-49.44-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.44" [] [[1.5, 0.0]] 3 true true false false 0.013015784565208074 0 2.67981217518042148357136879930 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/44/11"] "2-7e2-49.46-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.3246339017673209 0 0.52240080394910227366838007736 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/1"] "2-7e2-49.46-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.32974071444741865 0 1.00230215129235707707879419030 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/7"] "2-7e2-49.46-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.39571567068004976 0 3.73248746724695174802041802279 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/12"] "2-7e2-49.46-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.3741856748213888 0 3.83802520091011241088817741099 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/10"] "2-7e2-49.46-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.4084479479539141 0 4.29885478239962152245936743646 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/13"] "2-7e2-49.46-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.44863325632254203 0 1.22223753490711421829053738706 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/4"] "2-7e2-49.46-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.30324374237910007 0 1.22236438347175788601795884707 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/5"] "2-7e2-49.46-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.00392140179599096 0 1.75443105919305970367165537491 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/11"] "2-7e2-49.46-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.013378336685975663 0 1.81732020230986005125096780199 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/9"] "2-7e2-49.46-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.09477264825457603 0 1.83678206818288495501314946556 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/2"] "2-7e2-49.46-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false -0.04414539990040816 0 1.83995614252234754327946444364 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/6"] "2-7e2-49.46-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.25784825967608305 0 2.95943760696418905589226822476 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/3"] "2-7e2-49.46-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.46" [] [[1.5, 0.0]] 3 true true false false 0.2613070693333806 0 3.11533716805105634215422576064 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/46/8"] "2-7e2-49.8-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false -0.214965302950958 0 0.14969318129018350600145584471 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/1"] "2-7e2-49.8-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false -0.32969469101718635 0 0.16484426099511081004884171976 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/4"] "2-7e2-49.8-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false 0.25458975146689033 0 2.93083790453382595545766685292 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/5"] "2-7e2-49.8-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false 0.093074988006742 0 3.02473460961902403442396022372 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/8"] "2-7e2-49.8-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false 0.4816026096431002 0 3.38660880989070452280915997390 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/6"] "2-7e2-49.8-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false -0.4521505292961306 0 0.72235578538241783759695500101 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/10"] "2-7e2-49.8-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false -0.021084891427079623 0 0.75859643704366725058430304390 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/2"] "2-7e2-49.8-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false 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2.8910935902812915 2 49 "49.8" [] [[1.5, 0.0]] 3 true true false false 0.3616975266520253 0 2.74236424569100957374869731312 ["ModularForm/GL2/Q/holomorphic/49/4/e/a/8/3"] "2-7e2-49.9-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.05618599463746324 0 0.18895716600255199045973357810 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/1"] "2-7e2-49.9-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.18520910766600937 0 0.52613958249762078231607219909 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/5"] "2-7e2-49.9-c3-0-10" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.12058895130213032 0 2.90361612090558437027817805346 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/7"] "2-7e2-49.9-c3-0-11" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.30207999955485465 0 3.27916148642208675825817118949 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/10"] "2-7e2-49.9-c3-0-12" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.43250783024241407 0 3.50490027602114768388754010819 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/4"] "2-7e2-49.9-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.014890836389362195 0 1.14636149197339117469018796589 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/2"] "2-7e2-49.9-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.3406917814900678 0 1.22310679727384655307898739915 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/12"] "2-7e2-49.9-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.2610477186910829 0 2.03870232582630197455165991855 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/13"] "2-7e2-49.9-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.11469497192650975 0 2.10817315908257502736576400229 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/9"] "2-7e2-49.9-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.04443736777186531 0 2.21656056373622285817158083273 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/6"] "2-7e2-49.9-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false -0.158903046518286 0 2.62420566207152579711005252320 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/11"] "2-7e2-49.9-c3-0-8" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.42621583685961456 0 2.87882708185478696294269568621 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/3"] "2-7e2-49.9-c3-0-9" 1.700321613778197 2.8910935902812915 2 49 "49.9" [] [[1.5, 0.0]] 3 true true false false 0.25128225797713205 0 2.88593731369754770454810549875 ["ModularForm/GL2/Q/holomorphic/49/4/g/a/9/8"] "2-7e2-7.2-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.4282255948658145 0 0.71187396613198385835126615551 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/30/2"] "2-7e2-7.2-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false -0.14649371361272054 0 1.41194297365436200775023785024 ["ModularForm/GL2/Q/holomorphic/49/4/c/c/30/1"] "2-7e2-7.2-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false -0.31316038027938725 0 1.48202847095998175464269468530 ["ModularForm/GL2/Q/holomorphic/49/4/c/a/30/1"] "2-7e2-7.2-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.051601452080239404 0 2.07779906582429401816152577618 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/30/1"] "2-7e2-7.2-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.051601452080239404 0 2.25176300575461817891705979816 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/30/4"] "2-7e2-7.2-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.12632076055877442 0 2.38851112186511548991445905713 ["ModularForm/GL2/Q/holomorphic/49/4/c/b/30/1"] "2-7e2-7.2-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.3535062863872795 0 3.79768215730231864999750578192 ["ModularForm/GL2/Q/holomorphic/49/4/c/d/30/1"] "2-7e2-7.2-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "7.2" [] [[1.5, 0.0]] 3 true true false false 0.4282255948658145 0 4.20923120209997597634682106564 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/30/3"] "2-7e2-7.4-c3-0-0" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.051601452080239404 0 0.873325273054187424694295672963 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/18/1"] "2-7e2-7.4-c3-0-1" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.4282255948658145 0 1.18779700036515412361999396350 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/18/3"] "2-7e2-7.4-c3-0-2" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.3535062863872795 0 1.61408681898428496179850998824 ["ModularForm/GL2/Q/holomorphic/49/4/c/d/18/1"] "2-7e2-7.4-c3-0-3" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.12632076055877442 0 1.70256716586223987954185963424 ["ModularForm/GL2/Q/holomorphic/49/4/c/b/18/1"] "2-7e2-7.4-c3-0-4" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.051601452080239404 0 2.69872940460800571062035467619 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/18/4"] "2-7e2-7.4-c3-0-5" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false 0.14649371361272054 0 2.94662922883036392193526423579 ["ModularForm/GL2/Q/holomorphic/49/4/c/c/18/1"] "2-7e2-7.4-c3-0-6" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false 0.31316038027938725 0 3.15495962223867153442379067193 ["ModularForm/GL2/Q/holomorphic/49/4/c/a/18/1"] "2-7e2-7.4-c3-0-7" 1.700321613778197 2.8910935902812915 2 49 "7.4" [] [[1.5, 0.0]] 3 true true false false -0.4282255948658145 0 3.74036059967678764411312221647 ["ModularForm/GL2/Q/holomorphic/49/4/c/e/18/2"] # 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).