↑

Properties

Label 24-29e24-1.1-c1e12-0-10
Degree 2424
Conductor 1.252×10351.252\times 10^{35}
Sign 11
Analytic cond. 8.41157×1098.41157\times 10^{9}
Root an. cond. 2.591412.59141
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·8-s + 5·9-s + 4·10-s − 2·11-s − 6·12-s + 2·13-s − 4·15-s + 4·16-s − 24·17-s + 10·18-s − 12·19-s + 6·20-s − 4·22-s + 4·23-s − 8·24-s + 11·25-s + 4·26-s − 6·27-s − 8·30-s − 6·31-s + 4·33-s − 48·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 5/3·9-s + 1.26·10-s − 0.603·11-s − 1.73·12-s + 0.554·13-s − 1.03·15-s + 16-s − 5.82·17-s + 2.35·18-s − 2.75·19-s + 1.34·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s + 11/5·25-s + 0.784·26-s − 1.15·27-s − 1.46·30-s − 1.07·31-s + 0.696·33-s − 8.23·34-s + ⋯

Functional equation

Λ(s)=((2924)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((2924)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 292429^{24}
Sign: 11
Analytic conductor: 8.41157×1098.41157\times 10^{9}
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 2924, ( :[1/2]12), 1)(24,\ 29^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 24.0093461524.00934615
L(12)L(\frac12) ≈\approx 24.0093461524.00934615
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)
bad29 1 1
good2 1−pT+T2+T4+pT5−11T6+7pT7−7T8+p3T9−7T10−19pT11+93T12−19p2T13−7p2T14+p6T15−7p4T16+7p6T17−11p6T18+p8T19+p8T20+p10T22−p12T23+p12T24 1 - p T + T^{2} + T^{4} + p T^{5} - 11 T^{6} + 7 p T^{7} - 7 T^{8} + p^{3} T^{9} - 7 T^{10} - 19 p T^{11} + 93 T^{12} - 19 p^{2} T^{13} - 7 p^{2} T^{14} + p^{6} T^{15} - 7 p^{4} T^{16} + 7 p^{6} T^{17} - 11 p^{6} T^{18} + p^{8} T^{19} + p^{8} T^{20} + p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24}
3 1+2T−T2−2pT3−4T4−2T5−11T6−124T7−169T8+26p2T9+668T10+268T11+121T12+268pT13+668p2T14+26p5T15−169p4T16−124p5T17−11p6T18−2p7T19−4p8T20−2p10T21−p10T22+2p11T23+p12T24 1 + 2 T - T^{2} - 2 p T^{3} - 4 T^{4} - 2 T^{5} - 11 T^{6} - 124 T^{7} - 169 T^{8} + 26 p^{2} T^{9} + 668 T^{10} + 268 T^{11} + 121 T^{12} + 268 p T^{13} + 668 p^{2} T^{14} + 26 p^{5} T^{15} - 169 p^{4} T^{16} - 124 p^{5} T^{17} - 11 p^{6} T^{18} - 2 p^{7} T^{19} - 4 p^{8} T^{20} - 2 p^{10} T^{21} - p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
5 (1−T−4T2+9T3+11T4−56T5+T6−56pT7+11p2T8+9p3T9−4p4T10−p5T11+p6T12)2 ( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} - 56 T^{5} + T^{6} - 56 p T^{7} + 11 p^{2} T^{8} + 9 p^{3} T^{9} - 4 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2}
7 1−6T2−13T4+372T6−1595T8−8658T10+130103T12−8658p2T14−1595p4T16+372p6T18−13p8T20−6p10T22+p12T24 1 - 6 T^{2} - 13 T^{4} + 372 T^{6} - 1595 T^{8} - 8658 T^{10} + 130103 T^{12} - 8658 p^{2} T^{14} - 1595 p^{4} T^{16} + 372 p^{6} T^{18} - 13 p^{8} T^{20} - 6 p^{10} T^{22} + p^{12} T^{24}
11 1+2T−17T2−54T3+172T4+862T5−1019T6−9068T7+1415T8+66538T9+27164T10−214612T11+292953T12−214612pT13+27164p2T14+66538p3T15+1415p4T16−9068p5T17−1019p6T18+862p7T19+172p8T20−54p9T21−17p10T22+2p11T23+p12T24 1 + 2 T - 17 T^{2} - 54 T^{3} + 172 T^{4} + 862 T^{5} - 1019 T^{6} - 9068 T^{7} + 1415 T^{8} + 66538 T^{9} + 27164 T^{10} - 214612 T^{11} + 292953 T^{12} - 214612 p T^{13} + 27164 p^{2} T^{14} + 66538 p^{3} T^{15} + 1415 p^{4} T^{16} - 9068 p^{5} T^{17} - 1019 p^{6} T^{18} + 862 p^{7} T^{19} + 172 p^{8} T^{20} - 54 p^{9} T^{21} - 17 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
13 1−2T−15T2+42T3+84T4−238T5+323T6+21140T7−56425T8−256986T9+981820T10+813484T11−4064087T12+813484pT13+981820p2T14−256986p3T15−56425p4T16+21140p5T17+323p6T18−238p7T19+84p8T20+42p9T21−15p10T22−2p11T23+p12T24 1 - 2 T - 15 T^{2} + 42 T^{3} + 84 T^{4} - 238 T^{5} + 323 T^{6} + 21140 T^{7} - 56425 T^{8} - 256986 T^{9} + 981820 T^{10} + 813484 T^{11} - 4064087 T^{12} + 813484 p T^{13} + 981820 p^{2} T^{14} - 256986 p^{3} T^{15} - 56425 p^{4} T^{16} + 21140 p^{5} T^{17} + 323 p^{6} T^{18} - 238 p^{7} T^{19} + 84 p^{8} T^{20} + 42 p^{9} T^{21} - 15 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24}
17 (1+4T+30T2+4pT3+p2T4)6 ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{6}
19 (1+6T+17T2−12T3−395T4−2142T5−5347T6−2142pT7−395p2T8−12p3T9+17p4T10+6p5T11+p6T12)2 ( 1 + 6 T + 17 T^{2} - 12 T^{3} - 395 T^{4} - 2142 T^{5} - 5347 T^{6} - 2142 p T^{7} - 395 p^{2} T^{8} - 12 p^{3} T^{9} + 17 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2}
23 1−4T−2T2−12T3−77T4+2824T5−7748T6+17620T7−150091T8−497876T9+7170794T10−15234016T11+57064503T12−15234016pT13+7170794p2T14−497876p3T15−150091p4T16+17620p5T17−7748p6T18+2824p7T19−77p8T20−12p9T21−2p10T22−4p11T23+p12T24 1 - 4 T - 2 T^{2} - 12 T^{3} - 77 T^{4} + 2824 T^{5} - 7748 T^{6} + 17620 T^{7} - 150091 T^{8} - 497876 T^{9} + 7170794 T^{10} - 15234016 T^{11} + 57064503 T^{12} - 15234016 p T^{13} + 7170794 p^{2} T^{14} - 497876 p^{3} T^{15} - 150091 p^{4} T^{16} + 17620 p^{5} T^{17} - 7748 p^{6} T^{18} + 2824 p^{7} T^{19} - 77 p^{8} T^{20} - 12 p^{9} T^{21} - 2 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24}
31 1+6T+15T2+150T3+740T4−1686T5−12171T6+121764T7−38561T8−3431970T9+14562396T10+35258196T11−695550439T12+35258196pT13+14562396p2T14−3431970p3T15−38561p4T16+121764p5T17−12171p6T18−1686p7T19+740p8T20+150p9T21+15p10T22+6p11T23+p12T24 1 + 6 T + 15 T^{2} + 150 T^{3} + 740 T^{4} - 1686 T^{5} - 12171 T^{6} + 121764 T^{7} - 38561 T^{8} - 3431970 T^{9} + 14562396 T^{10} + 35258196 T^{11} - 695550439 T^{12} + 35258196 p T^{13} + 14562396 p^{2} T^{14} - 3431970 p^{3} T^{15} - 38561 p^{4} T^{16} + 121764 p^{5} T^{17} - 12171 p^{6} T^{18} - 1686 p^{7} T^{19} + 740 p^{8} T^{20} + 150 p^{9} T^{21} + 15 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24}
37 (1−4T−21T2+232T3−151T4−7980T5+37507T6−7980pT7−151p2T8+232p3T9−21p4T10−4p5T11+p6T12)2 ( 1 - 4 T - 21 T^{2} + 232 T^{3} - 151 T^{4} - 7980 T^{5} + 37507 T^{6} - 7980 p T^{7} - 151 p^{2} T^{8} + 232 p^{3} T^{9} - 21 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2}
41 (1−8T+26T2−8pT3+p2T4)6 ( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{6}
43 1+10T−9T2−750T3−4068T4+18710T5+324653T6+1377620T7−6043945T8−105594030T9−405057092T10+2313062620T11+33051670633T12+2313062620pT13−405057092p2T14−105594030p3T15−6043945p4T16+1377620p5T17+324653p6T18+18710p7T19−4068p8T20−750p9T21−9p10T22+10p11T23+p12T24 1 + 10 T - 9 T^{2} - 750 T^{3} - 4068 T^{4} + 18710 T^{5} + 324653 T^{6} + 1377620 T^{7} - 6043945 T^{8} - 105594030 T^{9} - 405057092 T^{10} + 2313062620 T^{11} + 33051670633 T^{12} + 2313062620 p T^{13} - 405057092 p^{2} T^{14} - 105594030 p^{3} T^{15} - 6043945 p^{4} T^{16} + 1377620 p^{5} T^{17} + 324653 p^{6} T^{18} + 18710 p^{7} T^{19} - 4068 p^{8} T^{20} - 750 p^{9} T^{21} - 9 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24}
47 1+2T−73T2−206T3+3188T4+10958T5−81667T6−1120740T7−1965361T8+50483338T9+327352316T10−941541668T11−18002295447T12−941541668pT13+327352316p2T14+50483338p3T15−1965361p4T16−1120740p5T17−81667p6T18+10958p7T19+3188p8T20−206p9T21−73p10T22+2p11T23+p12T24 1 + 2 T - 73 T^{2} - 206 T^{3} + 3188 T^{4} + 10958 T^{5} - 81667 T^{6} - 1120740 T^{7} - 1965361 T^{8} + 50483338 T^{9} + 327352316 T^{10} - 941541668 T^{11} - 18002295447 T^{12} - 941541668 p T^{13} + 327352316 p^{2} T^{14} + 50483338 p^{3} T^{15} - 1965361 p^{4} T^{16} - 1120740 p^{5} T^{17} - 81667 p^{6} T^{18} + 10958 p^{7} T^{19} + 3188 p^{8} T^{20} - 206 p^{9} T^{21} - 73 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
53 1+2T−31T2−26T3−1564T4−11122T5+116819T6−276804T7−1427929T8+50456794T9−206594692T10−1552995212T11+13484296521T12−1552995212pT13−206594692p2T14+50456794p3T15−1427929p4T16−276804p5T17+116819p6T18−11122p7T19−1564p8T20−26p9T21−31p10T22+2p11T23+p12T24 1 + 2 T - 31 T^{2} - 26 T^{3} - 1564 T^{4} - 11122 T^{5} + 116819 T^{6} - 276804 T^{7} - 1427929 T^{8} + 50456794 T^{9} - 206594692 T^{10} - 1552995212 T^{11} + 13484296521 T^{12} - 1552995212 p T^{13} - 206594692 p^{2} T^{14} + 50456794 p^{3} T^{15} - 1427929 p^{4} T^{16} - 276804 p^{5} T^{17} + 116819 p^{6} T^{18} - 11122 p^{7} T^{19} - 1564 p^{8} T^{20} - 26 p^{9} T^{21} - 31 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24}
59 (1−4T+90T2−4pT3+p2T4)6 ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{6}
61 1−4T−102T2+636T3+6747T4−62264T5−322732T6+3045052T7+15988997T8−112841172T9−977443874T10+1993080032T11+75404245135T12+1993080032pT13−977443874p2T14−112841172p3T15+15988997p4T16+3045052p5T17−322732p6T18−62264p7T19+6747p8T20+636p9T21−102p10T22−4p11T23+p12T24 1 - 4 T - 102 T^{2} + 636 T^{3} + 6747 T^{4} - 62264 T^{5} - 322732 T^{6} + 3045052 T^{7} + 15988997 T^{8} - 112841172 T^{9} - 977443874 T^{10} + 1993080032 T^{11} + 75404245135 T^{12} + 1993080032 p T^{13} - 977443874 p^{2} T^{14} - 112841172 p^{3} T^{15} + 15988997 p^{4} T^{16} + 3045052 p^{5} T^{17} - 322732 p^{6} T^{18} - 62264 p^{7} T^{19} + 6747 p^{8} T^{20} + 636 p^{9} T^{21} - 102 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24}
67 1−102T2+5915T4−145452T6−11716331T8+1847999790T10−135901368721T12+1847999790p2T14−11716331p4T16−145452p6T18+5915p8T20−102p10T22+p12T24 1 - 102 T^{2} + 5915 T^{4} - 145452 T^{6} - 11716331 T^{8} + 1847999790 T^{10} - 135901368721 T^{12} + 1847999790 p^{2} T^{14} - 11716331 p^{4} T^{16} - 145452 p^{6} T^{18} + 5915 p^{8} T^{20} - 102 p^{10} T^{22} + p^{12} T^{24}
71 1−12T−26T2+1500T3−8397T4−71592T5+1139660T6−3794628T7−44881003T8+573563748T9−1764997470T10−18958533216T11+265120791319T12−18958533216pT13−1764997470p2T14+573563748p3T15−44881003p4T16−3794628p5T17+1139660p6T18−71592p7T19−8397p8T20+1500p9T21−26p10T22−12p11T23+p12T24 1 - 12 T - 26 T^{2} + 1500 T^{3} - 8397 T^{4} - 71592 T^{5} + 1139660 T^{6} - 3794628 T^{7} - 44881003 T^{8} + 573563748 T^{9} - 1764997470 T^{10} - 18958533216 T^{11} + 265120791319 T^{12} - 18958533216 p T^{13} - 1764997470 p^{2} T^{14} + 573563748 p^{3} T^{15} - 44881003 p^{4} T^{16} - 3794628 p^{5} T^{17} + 1139660 p^{6} T^{18} - 71592 p^{7} T^{19} - 8397 p^{8} T^{20} + 1500 p^{9} T^{21} - 26 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
73 (1+4T−57T2−520T3+2081T4+46284T5+33223T6+46284pT7+2081p2T8−520p3T9−57p4T10+4p5T11+p6T12)2 ( 1 + 4 T - 57 T^{2} - 520 T^{3} + 2081 T^{4} + 46284 T^{5} + 33223 T^{6} + 46284 p T^{7} + 2081 p^{2} T^{8} - 520 p^{3} T^{9} - 57 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2}
79 1−2T−153T2+462T3+17172T4−70222T5−1673683T6+3051524T7+160589807T8−97571466T9−15054141956T10+1009253284T11+1374657086665T12+1009253284pT13−15054141956p2T14−97571466p3T15+160589807p4T16+3051524p5T17−1673683p6T18−70222p7T19+17172p8T20+462p9T21−153p10T22−2p11T23+p12T24 1 - 2 T - 153 T^{2} + 462 T^{3} + 17172 T^{4} - 70222 T^{5} - 1673683 T^{6} + 3051524 T^{7} + 160589807 T^{8} - 97571466 T^{9} - 15054141956 T^{10} + 1009253284 T^{11} + 1374657086665 T^{12} + 1009253284 p T^{13} - 15054141956 p^{2} T^{14} - 97571466 p^{3} T^{15} + 160589807 p^{4} T^{16} + 3051524 p^{5} T^{17} - 1673683 p^{6} T^{18} - 70222 p^{7} T^{19} + 17172 p^{8} T^{20} + 462 p^{9} T^{21} - 153 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24}
83 1+4T−122T2−708T3+8443T4+63416T5−306068T6−11486980T7−40820251T8+883434836T9+7461759074T10−26485489184T11−561153634257T12−26485489184pT13+7461759074p2T14+883434836p3T15−40820251p4T16−11486980p5T17−306068p6T18+63416p7T19+8443p8T20−708p9T21−122p10T22+4p11T23+p12T24 1 + 4 T - 122 T^{2} - 708 T^{3} + 8443 T^{4} + 63416 T^{5} - 306068 T^{6} - 11486980 T^{7} - 40820251 T^{8} + 883434836 T^{9} + 7461759074 T^{10} - 26485489184 T^{11} - 561153634257 T^{12} - 26485489184 p T^{13} + 7461759074 p^{2} T^{14} + 883434836 p^{3} T^{15} - 40820251 p^{4} T^{16} - 11486980 p^{5} T^{17} - 306068 p^{6} T^{18} + 63416 p^{7} T^{19} + 8443 p^{8} T^{20} - 708 p^{9} T^{21} - 122 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24}
89 1−8T−58T2+728T3−973T4+23632T5−129268T6+18797016T7−143724283T8−1238591944T9+15083554514T10−16119693184T11+309081404535T12−16119693184pT13+15083554514p2T14−1238591944p3T15−143724283p4T16+18797016p5T17−129268p6T18+23632p7T19−973p8T20+728p9T21−58p10T22−8p11T23+p12T24 1 - 8 T - 58 T^{2} + 728 T^{3} - 973 T^{4} + 23632 T^{5} - 129268 T^{6} + 18797016 T^{7} - 143724283 T^{8} - 1238591944 T^{9} + 15083554514 T^{10} - 16119693184 T^{11} + 309081404535 T^{12} - 16119693184 p T^{13} + 15083554514 p^{2} T^{14} - 1238591944 p^{3} T^{15} - 143724283 p^{4} T^{16} + 18797016 p^{5} T^{17} - 129268 p^{6} T^{18} + 23632 p^{7} T^{19} - 973 p^{8} T^{20} + 728 p^{9} T^{21} - 58 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
97 1−8T−74T2+920T3−349T4+8528T5−37716T6+26604056T7−210961627T8−2034639048T9+25099939298T10−6630666112T11+153071555463T12−6630666112pT13+25099939298p2T14−2034639048p3T15−210961627p4T16+26604056p5T17−37716p6T18+8528p7T19−349p8T20+920p9T21−74p10T22−8p11T23+p12T24 1 - 8 T - 74 T^{2} + 920 T^{3} - 349 T^{4} + 8528 T^{5} - 37716 T^{6} + 26604056 T^{7} - 210961627 T^{8} - 2034639048 T^{9} + 25099939298 T^{10} - 6630666112 T^{11} + 153071555463 T^{12} - 6630666112 p T^{13} + 25099939298 p^{2} T^{14} - 2034639048 p^{3} T^{15} - 210961627 p^{4} T^{16} + 26604056 p^{5} T^{17} - 37716 p^{6} T^{18} + 8528 p^{7} T^{19} - 349 p^{8} T^{20} + 920 p^{9} T^{21} - 74 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24}
show more
show less
   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.20712174088636956492492552182, −3.08809539918140425934697908220, −3.02964510995801303773866080692, −3.02672991394293779209960180779, −2.79757472611951062168404157234, −2.79530424399108927083643204803, −2.65832785419631649050235224406, −2.56996292500046844146149735898, −2.45331475648531914849479260377, −2.36580081380291123328599448126, −2.29504996597483172484529119019, −2.02103877084818031845182020650, −1.97417881063653645072917151907, −1.96905803319852487910584569694, −1.96607414579134336926880466710, −1.80912186519909524031343718907, −1.80488654398499356800401042462, −1.58835952916660353698575374839, −1.28218722185610260080672308533, −0.966415505681229254605213773632, −0.792948669759499669336840440010, −0.78844514174138766580760551541, −0.64211670183940325012502179823, −0.56054857004110585455592721097, −0.31125417577498656784524517216, 0.31125417577498656784524517216, 0.56054857004110585455592721097, 0.64211670183940325012502179823, 0.78844514174138766580760551541, 0.792948669759499669336840440010, 0.966415505681229254605213773632, 1.28218722185610260080672308533, 1.58835952916660353698575374839, 1.80488654398499356800401042462, 1.80912186519909524031343718907, 1.96607414579134336926880466710, 1.96905803319852487910584569694, 1.97417881063653645072917151907, 2.02103877084818031845182020650, 2.29504996597483172484529119019, 2.36580081380291123328599448126, 2.45331475648531914849479260377, 2.56996292500046844146149735898, 2.65832785419631649050235224406, 2.79530424399108927083643204803, 2.79757472611951062168404157234, 3.02672991394293779209960180779, 3.02964510995801303773866080692, 3.08809539918140425934697908220, 3.20712174088636956492492552182

Graph of the ZZ-function along the critical line

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