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Properties

Label 20-63e10-1.1-c1e10-0-0
Degree 2020
Conductor 9.849×10179.849\times 10^{17}
Sign 11
Analytic cond. 0.001037950.00103795
Root an. cond. 0.7092650.709265
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 3-s + 2·4-s + 4·5-s + 4·6-s − 4·7-s + 14·8-s + 6·9-s − 16·10-s + 4·11-s − 2·12-s − 8·13-s + 16·14-s − 4·15-s − 27·16-s + 12·17-s − 24·18-s + 19-s + 8·20-s + 4·21-s − 16·22-s + 3·23-s − 14·24-s + 20·25-s + 32·26-s − 8·27-s − 8·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 4-s + 1.78·5-s + 1.63·6-s − 1.51·7-s + 4.94·8-s + 2·9-s − 5.05·10-s + 1.20·11-s − 0.577·12-s − 2.21·13-s + 4.27·14-s − 1.03·15-s − 6.75·16-s + 2.91·17-s − 5.65·18-s + 0.229·19-s + 1.78·20-s + 0.872·21-s − 3.41·22-s + 0.625·23-s − 2.85·24-s + 4·25-s + 6.27·26-s − 1.53·27-s − 1.51·28-s + ⋯

Functional equation

Λ(s)=((320⋅710)s/2 ΓC(s)10 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((320⋅710)s/2 ΓC(s+1/2)10 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2020
Conductor: 320⋅7103^{20} \cdot 7^{10}
Sign: 11
Analytic conductor: 0.001037950.00103795
Root analytic conductor: 0.7092650.709265
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (20, 320⋅710, ( :[1/2]10), 1)(20,\ 3^{20} \cdot 7^{10} ,\ ( \ : [1/2]^{10} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.044917250630.04491725063
L(12)L(\frac12) ≈\approx 0.044917250630.04491725063
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+T−5T2−pT3+p2T4+p2T5+p3T6−p3T7−5p3T8+p4T9+p5T10 1 + T - 5 T^{2} - p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} - p^{3} T^{7} - 5 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10}
7 1+4T+12T2+47T3+146T4+309T5+146pT6+47p2T7+12p3T8+4p4T9+p5T10 1 + 4 T + 12 T^{2} + 47 T^{3} + 146 T^{4} + 309 T^{5} + 146 p T^{6} + 47 p^{2} T^{7} + 12 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10}
good2 (1+pT+5T2+7T3+13T4+15T5+13pT6+7p2T7+5p3T8+p5T9+p5T10)2 ( 1 + p T + 5 T^{2} + 7 T^{3} + 13 T^{4} + 15 T^{5} + 13 p T^{6} + 7 p^{2} T^{7} + 5 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} )^{2}
5 1−4T−4T2+44T3−41T4−119T5+222T6−456T7+1623T8+2021T9−16541T10+2021pT11+1623p2T12−456p3T13+222p4T14−119p5T15−41p6T16+44p7T17−4p8T18−4p9T19+p10T20 1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 2021 p T^{11} + 1623 p^{2} T^{12} - 456 p^{3} T^{13} + 222 p^{4} T^{14} - 119 p^{5} T^{15} - 41 p^{6} T^{16} + 44 p^{7} T^{17} - 4 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
11 1−4T−31T2+134T3+607T4−2492T5−8385T6+27495T7+98940T8−135733T9−1043873T10−135733pT11+98940p2T12+27495p3T13−8385p4T14−2492p5T15+607p6T16+134p7T17−31p8T18−4p9T19+p10T20 1 - 4 T - 31 T^{2} + 134 T^{3} + 607 T^{4} - 2492 T^{5} - 8385 T^{6} + 27495 T^{7} + 98940 T^{8} - 135733 T^{9} - 1043873 T^{10} - 135733 p T^{11} + 98940 p^{2} T^{12} + 27495 p^{3} T^{13} - 8385 p^{4} T^{14} - 2492 p^{5} T^{15} + 607 p^{6} T^{16} + 134 p^{7} T^{17} - 31 p^{8} T^{18} - 4 p^{9} T^{19} + p^{10} T^{20}
13 1+8T−14T2−14pT3+686T4+4429T5−12871T6−3323pT7+305249T8+358672T9−3841969T10+358672pT11+305249p2T12−3323p4T13−12871p4T14+4429p5T15+686p6T16−14p8T17−14p8T18+8p9T19+p10T20 1 + 8 T - 14 T^{2} - 14 p T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 3323 p T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 358672 p T^{11} + 305249 p^{2} T^{12} - 3323 p^{4} T^{13} - 12871 p^{4} T^{14} + 4429 p^{5} T^{15} + 686 p^{6} T^{16} - 14 p^{8} T^{17} - 14 p^{8} T^{18} + 8 p^{9} T^{19} + p^{10} T^{20}
17 1−12T+14T2+192T3+1185T4−11847T5−6180T6+65736T7+1002861T8−2436261T9−7749777T10−2436261pT11+1002861p2T12+65736p3T13−6180p4T14−11847p5T15+1185p6T16+192p7T17+14p8T18−12p9T19+p10T20 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 2436261 p T^{11} + 1002861 p^{2} T^{12} + 65736 p^{3} T^{13} - 6180 p^{4} T^{14} - 11847 p^{5} T^{15} + 1185 p^{6} T^{16} + 192 p^{7} T^{17} + 14 p^{8} T^{18} - 12 p^{9} T^{19} + p^{10} T^{20}
19 1−T−53T2+10pT3+1262T4−7007T5−13111T6+116110T7+67964T8−721616T9−440023T10−721616pT11+67964p2T12+116110p3T13−13111p4T14−7007p5T15+1262p6T16+10p8T17−53p8T18−p9T19+p10T20 1 - T - 53 T^{2} + 10 p T^{3} + 1262 T^{4} - 7007 T^{5} - 13111 T^{6} + 116110 T^{7} + 67964 T^{8} - 721616 T^{9} - 440023 T^{10} - 721616 p T^{11} + 67964 p^{2} T^{12} + 116110 p^{3} T^{13} - 13111 p^{4} T^{14} - 7007 p^{5} T^{15} + 1262 p^{6} T^{16} + 10 p^{8} T^{17} - 53 p^{8} T^{18} - p^{9} T^{19} + p^{10} T^{20}
23 1−3T−43T2+294T3+6T4−5127T5+21792T6−135027T7+502362T8+3271749T9−33095343T10+3271749pT11+502362p2T12−135027p3T13+21792p4T14−5127p5T15+6p6T16+294p7T17−43p8T18−3p9T19+p10T20 1 - 3 T - 43 T^{2} + 294 T^{3} + 6 T^{4} - 5127 T^{5} + 21792 T^{6} - 135027 T^{7} + 502362 T^{8} + 3271749 T^{9} - 33095343 T^{10} + 3271749 p T^{11} + 502362 p^{2} T^{12} - 135027 p^{3} T^{13} + 21792 p^{4} T^{14} - 5127 p^{5} T^{15} + 6 p^{6} T^{16} + 294 p^{7} T^{17} - 43 p^{8} T^{18} - 3 p^{9} T^{19} + p^{10} T^{20}
29 1−7T−76T2+419T3+4561T4−15146T5−199563T6+341373T7+6918636T8−2570041T9−219913241T10−2570041pT11+6918636p2T12+341373p3T13−199563p4T14−15146p5T15+4561p6T16+419p7T17−76p8T18−7p9T19+p10T20 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 2570041 p T^{11} + 6918636 p^{2} T^{12} + 341373 p^{3} T^{13} - 199563 p^{4} T^{14} - 15146 p^{5} T^{15} + 4561 p^{6} T^{16} + 419 p^{7} T^{17} - 76 p^{8} T^{18} - 7 p^{9} T^{19} + p^{10} T^{20}
31 (1−3T+134T2−308T3+250pT4−13615T5+250p2T6−308p2T7+134p3T8−3p4T9+p5T10)2 ( 1 - 3 T + 134 T^{2} - 308 T^{3} + 250 p T^{4} - 13615 T^{5} + 250 p^{2} T^{6} - 308 p^{2} T^{7} + 134 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2}
37 1−89T2+560T3+4503T4−45352T5+27130T6+2296536T7−9801827T8−33131096T9+610977105T10−33131096pT11−9801827p2T12+2296536p3T13+27130p4T14−45352p5T15+4503p6T16+560p7T17−89p8T18+p10T20 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 33131096 p T^{11} - 9801827 p^{2} T^{12} + 2296536 p^{3} T^{13} + 27130 p^{4} T^{14} - 45352 p^{5} T^{15} + 4503 p^{6} T^{16} + 560 p^{7} T^{17} - 89 p^{8} T^{18} + p^{10} T^{20}
41 1−5T−136T2+733T3+10507T4−54412T5−554055T6+2345451T7+23706084T8−41392439T9−952045937T10−41392439pT11+23706084p2T12+2345451p3T13−554055p4T14−54412p5T15+10507p6T16+733p7T17−136p8T18−5p9T19+p10T20 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 41392439 p T^{11} + 23706084 p^{2} T^{12} + 2345451 p^{3} T^{13} - 554055 p^{4} T^{14} - 54412 p^{5} T^{15} + 10507 p^{6} T^{16} + 733 p^{7} T^{17} - 136 p^{8} T^{18} - 5 p^{9} T^{19} + p^{10} T^{20}
43 1+7T−77T2−66T3+7014T4−3843T5−95427T6+1632678T7−3708600T8−15416324T9+670279801T10−15416324pT11−3708600p2T12+1632678p3T13−95427p4T14−3843p5T15+7014p6T16−66p7T17−77p8T18+7p9T19+p10T20 1 + 7 T - 77 T^{2} - 66 T^{3} + 7014 T^{4} - 3843 T^{5} - 95427 T^{6} + 1632678 T^{7} - 3708600 T^{8} - 15416324 T^{9} + 670279801 T^{10} - 15416324 p T^{11} - 3708600 p^{2} T^{12} + 1632678 p^{3} T^{13} - 95427 p^{4} T^{14} - 3843 p^{5} T^{15} + 7014 p^{6} T^{16} - 66 p^{7} T^{17} - 77 p^{8} T^{18} + 7 p^{9} T^{19} + p^{10} T^{20}
47 (1+27T+448T2+5169T3+48091T4+359985T5+48091pT6+5169p2T7+448p3T8+27p4T9+p5T10)2 ( 1 + 27 T + 448 T^{2} + 5169 T^{3} + 48091 T^{4} + 359985 T^{5} + 48091 p T^{6} + 5169 p^{2} T^{7} + 448 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} )^{2}
53 1+21T+41T2−924T3+12966T4+177027T5−601755T6−3783942T7+110973258T8+340111866T9−4044436041T10+340111866pT11+110973258p2T12−3783942p3T13−601755p4T14+177027p5T15+12966p6T16−924p7T17+41p8T18+21p9T19+p10T20 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 340111866 p T^{11} + 110973258 p^{2} T^{12} - 3783942 p^{3} T^{13} - 601755 p^{4} T^{14} + 177027 p^{5} T^{15} + 12966 p^{6} T^{16} - 924 p^{7} T^{17} + 41 p^{8} T^{18} + 21 p^{9} T^{19} + p^{10} T^{20}
59 (1+30T+601T2+8193T3+88864T4+752289T5+88864pT6+8193p2T7+601p3T8+30p4T9+p5T10)2 ( 1 + 30 T + 601 T^{2} + 8193 T^{3} + 88864 T^{4} + 752289 T^{5} + 88864 p T^{6} + 8193 p^{2} T^{7} + 601 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} )^{2}
61 (1−14T+339T2−3409T3+43418T4−311709T5+43418pT6−3409p2T7+339p3T8−14p4T9+p5T10)2 ( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 43418 p T^{6} - 3409 p^{2} T^{7} + 339 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{2}
67 (1−2T+132T2−196T3+10871T4−15429T5+10871pT6−196p2T7+132p3T8−2p4T9+p5T10)2 ( 1 - 2 T + 132 T^{2} - 196 T^{3} + 10871 T^{4} - 15429 T^{5} + 10871 p T^{6} - 196 p^{2} T^{7} + 132 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2}
71 (1+3T+187T2+285T3+15679T4+10143T5+15679pT6+285p2T7+187p3T8+3p4T9+p5T10)2 ( 1 + 3 T + 187 T^{2} + 285 T^{3} + 15679 T^{4} + 10143 T^{5} + 15679 p T^{6} + 285 p^{2} T^{7} + 187 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2}
73 1−15T−134T2+2501T3+16563T4−235276T5−2002535T6+9021201T7+288508378T8−238799411T9−25271949561T10−238799411pT11+288508378p2T12+9021201p3T13−2002535p4T14−235276p5T15+16563p6T16+2501p7T17−134p8T18−15p9T19+p10T20 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 238799411 p T^{11} + 288508378 p^{2} T^{12} + 9021201 p^{3} T^{13} - 2002535 p^{4} T^{14} - 235276 p^{5} T^{15} + 16563 p^{6} T^{16} + 2501 p^{7} T^{17} - 134 p^{8} T^{18} - 15 p^{9} T^{19} + p^{10} T^{20}
79 (1−4T+300T2−1488T3+39873T4−184983T5+39873pT6−1488p2T7+300p3T8−4p4T9+p5T10)2 ( 1 - 4 T + 300 T^{2} - 1488 T^{3} + 39873 T^{4} - 184983 T^{5} + 39873 p T^{6} - 1488 p^{2} T^{7} + 300 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2}
83 1−9T−148T2−297T3+24654T4+118125T5−807174T6−21382137T7−37648479T8+452536146T9+15509586612T10+452536146pT11−37648479p2T12−21382137p3T13−807174p4T14+118125p5T15+24654p6T16−297p7T17−148p8T18−9p9T19+p10T20 1 - 9 T - 148 T^{2} - 297 T^{3} + 24654 T^{4} + 118125 T^{5} - 807174 T^{6} - 21382137 T^{7} - 37648479 T^{8} + 452536146 T^{9} + 15509586612 T^{10} + 452536146 p T^{11} - 37648479 p^{2} T^{12} - 21382137 p^{3} T^{13} - 807174 p^{4} T^{14} + 118125 p^{5} T^{15} + 24654 p^{6} T^{16} - 297 p^{7} T^{17} - 148 p^{8} T^{18} - 9 p^{9} T^{19} + p^{10} T^{20}
89 1−28T+104T2+1736T3+31273T4−611939T5−1780638T6+18973932T7+740914101T8−3271180573T9−40614588329T10−3271180573pT11+740914101p2T12+18973932p3T13−1780638p4T14−611939p5T15+31273p6T16+1736p7T17+104p8T18−28p9T19+p10T20 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 3271180573 p T^{11} + 740914101 p^{2} T^{12} + 18973932 p^{3} T^{13} - 1780638 p^{4} T^{14} - 611939 p^{5} T^{15} + 31273 p^{6} T^{16} + 1736 p^{7} T^{17} + 104 p^{8} T^{18} - 28 p^{9} T^{19} + p^{10} T^{20}
97 1+12T−197T2−1534T3+27813T4+14090T5−4545035T6−6881349T7+472663750T8+908843245T9−38512186359T10+908843245pT11+472663750p2T12−6881349p3T13−4545035p4T14+14090p5T15+27813p6T16−1534p7T17−197p8T18+12p9T19+p10T20 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 908843245 p T^{11} + 472663750 p^{2} T^{12} - 6881349 p^{3} T^{13} - 4545035 p^{4} T^{14} + 14090 p^{5} T^{15} + 27813 p^{6} T^{16} - 1534 p^{7} T^{17} - 197 p^{8} T^{18} + 12 p^{9} T^{19} + p^{10} T^{20}
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   L(s)=∏p ∏j=120(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.53207955415737755395834729244, −6.25303642141925168021465339098, −6.07013110388134911569095207318, −5.95279916044332521466658862994, −5.94026064560954979798465437727, −5.91132333595949228298131116926, −5.10695350308937058297288025336, −5.02725687679474812737482165602, −4.95428750136026086516854026604, −4.91231592321559048897814000150, −4.89905415517488434285814136983, −4.82158448379452796949920558183, −4.49149929585291352046660725836, −4.38903082395480488332444143863, −4.38322358029979438503815601339, −3.51326071174946722430497290653, −3.43742014098811917873730442880, −3.31792143020839315790164686836, −3.29039209831810741951829606088, −3.13888849535205387440684217146, −2.71428091851818167901520909891, −2.15573284589757776053100995816, −1.63497820121561289417728828021, −1.58907409519659394128148289584, −1.04350541068247671363544768239, 1.04350541068247671363544768239, 1.58907409519659394128148289584, 1.63497820121561289417728828021, 2.15573284589757776053100995816, 2.71428091851818167901520909891, 3.13888849535205387440684217146, 3.29039209831810741951829606088, 3.31792143020839315790164686836, 3.43742014098811917873730442880, 3.51326071174946722430497290653, 4.38322358029979438503815601339, 4.38903082395480488332444143863, 4.49149929585291352046660725836, 4.82158448379452796949920558183, 4.89905415517488434285814136983, 4.91231592321559048897814000150, 4.95428750136026086516854026604, 5.02725687679474812737482165602, 5.10695350308937058297288025336, 5.91132333595949228298131116926, 5.94026064560954979798465437727, 5.95279916044332521466658862994, 6.07013110388134911569095207318, 6.25303642141925168021465339098, 6.53207955415737755395834729244

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.