Dirichlet series
| L(s) = 1 | − 3-s + 3·5-s − 7-s + 4·11-s + 18·13-s − 3·15-s + 3·17-s + 4·19-s + 21-s − 18·23-s + 3·25-s + 4·27-s + 15·29-s − 8·31-s − 4·33-s − 3·35-s + 6·37-s − 18·39-s + 2·41-s − 36·43-s − 16·47-s + 3·49-s − 3·51-s + 19·53-s + 12·55-s − 4·57-s + 46·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.377·7-s + 1.20·11-s + 4.99·13-s − 0.774·15-s + 0.727·17-s + 0.917·19-s + 0.218·21-s − 3.75·23-s + 3/5·25-s + 0.769·27-s + 2.78·29-s − 1.43·31-s − 0.696·33-s − 0.507·35-s + 0.986·37-s − 2.88·39-s + 0.312·41-s − 5.48·43-s − 2.33·47-s + 3/7·49-s − 0.420·51-s + 2.60·53-s + 1.61·55-s − 0.529·57-s + 5.98·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 5^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.53799\times 10^{6}\) |
| Root analytic conductor: | \(1.87441\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 5^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.233550067\) |
| \(L(\frac12)\) | \(\approx\) | \(1.233550067\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) | |
| 11 | \( 1 - 4 T + 12 T^{2} - 3 T^{3} + 8 T^{4} + 122 T^{5} + 523 T^{6} + 122 p T^{7} + 8 p^{2} T^{8} - 3 p^{3} T^{9} + 12 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + T + T^{2} - p T^{3} - 7 T^{4} - 2 p^{2} T^{5} - p^{2} T^{6} + 56 T^{7} + 8 p T^{8} + 28 T^{9} + 71 T^{10} + 140 T^{11} - 335 T^{12} + 140 p T^{13} + 71 p^{2} T^{14} + 28 p^{3} T^{15} + 8 p^{5} T^{16} + 56 p^{5} T^{17} - p^{8} T^{18} - 2 p^{9} T^{19} - 7 p^{8} T^{20} - p^{10} T^{21} + p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 + T - 2 T^{2} - 22 T^{3} + 41 T^{4} + 468 T^{5} + 167 T^{6} - 1608 T^{7} - 5048 T^{8} + 23084 T^{9} + 90989 T^{10} - 50135 T^{11} - 342845 T^{12} - 50135 p T^{13} + 90989 p^{2} T^{14} + 23084 p^{3} T^{15} - 5048 p^{4} T^{16} - 1608 p^{5} T^{17} + 167 p^{6} T^{18} + 468 p^{7} T^{19} + 41 p^{8} T^{20} - 22 p^{9} T^{21} - 2 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 18 T + 146 T^{2} - 59 p T^{3} + 3353 T^{4} - 12748 T^{5} + 34986 T^{6} - 64446 T^{7} + 149199 T^{8} - 565472 T^{9} + 2330421 T^{10} - 14523541 T^{11} + 70061385 T^{12} - 14523541 p T^{13} + 2330421 p^{2} T^{14} - 565472 p^{3} T^{15} + 149199 p^{4} T^{16} - 64446 p^{5} T^{17} + 34986 p^{6} T^{18} - 12748 p^{7} T^{19} + 3353 p^{8} T^{20} - 59 p^{10} T^{21} + 146 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 3 T - 5 T^{2} + 100 T^{3} + 103 T^{4} - 2024 T^{5} + 633 T^{6} + 51620 T^{7} - 124988 T^{8} - 512836 T^{9} + 3579629 T^{10} + 3468039 T^{11} - 48726881 T^{12} + 3468039 p T^{13} + 3579629 p^{2} T^{14} - 512836 p^{3} T^{15} - 124988 p^{4} T^{16} + 51620 p^{5} T^{17} + 633 p^{6} T^{18} - 2024 p^{7} T^{19} + 103 p^{8} T^{20} + 100 p^{9} T^{21} - 5 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 4 T - 15 T^{2} - 51 T^{3} + 575 T^{4} + 5927 T^{5} - 21051 T^{6} - 75493 T^{7} - 334673 T^{8} + 2660406 T^{9} + 14597891 T^{10} - 44115627 T^{11} - 162320840 T^{12} - 44115627 p T^{13} + 14597891 p^{2} T^{14} + 2660406 p^{3} T^{15} - 334673 p^{4} T^{16} - 75493 p^{5} T^{17} - 21051 p^{6} T^{18} + 5927 p^{7} T^{19} + 575 p^{8} T^{20} - 51 p^{9} T^{21} - 15 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 9 T + 90 T^{2} + 583 T^{3} + 3944 T^{4} + 21650 T^{5} + 116494 T^{6} + 21650 p T^{7} + 3944 p^{2} T^{8} + 583 p^{3} T^{9} + 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 15 T + 76 T^{2} - 65 T^{3} - 475 T^{4} - 3145 T^{5} + 45255 T^{6} - 126070 T^{7} + 3194 p T^{8} - 5468375 T^{9} + 54266825 T^{10} - 257834160 T^{11} + 1157424369 T^{12} - 257834160 p T^{13} + 54266825 p^{2} T^{14} - 5468375 p^{3} T^{15} + 3194 p^{5} T^{16} - 126070 p^{5} T^{17} + 45255 p^{6} T^{18} - 3145 p^{7} T^{19} - 475 p^{8} T^{20} - 65 p^{9} T^{21} + 76 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 8 T - 12 T^{2} - 179 T^{3} + 879 T^{4} + 20928 T^{5} + 73290 T^{6} - 432638 T^{7} - 1361273 T^{8} + 20389752 T^{9} + 172083725 T^{10} + 2770013 p T^{11} - 132696187 p T^{12} + 2770013 p^{2} T^{13} + 172083725 p^{2} T^{14} + 20389752 p^{3} T^{15} - 1361273 p^{4} T^{16} - 432638 p^{5} T^{17} + 73290 p^{6} T^{18} + 20928 p^{7} T^{19} + 879 p^{8} T^{20} - 179 p^{9} T^{21} - 12 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 6 T - 104 T^{2} + 10 p T^{3} + 5594 T^{4} + 6764 T^{5} - 136053 T^{6} - 54442 p T^{7} - 2461520 T^{8} + 107282676 T^{9} + 411948722 T^{10} - 1966035362 T^{11} - 20003487727 T^{12} - 1966035362 p T^{13} + 411948722 p^{2} T^{14} + 107282676 p^{3} T^{15} - 2461520 p^{4} T^{16} - 54442 p^{6} T^{17} - 136053 p^{6} T^{18} + 6764 p^{7} T^{19} + 5594 p^{8} T^{20} + 10 p^{10} T^{21} - 104 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 2 T - 34 T^{2} + 130 T^{3} + 1002 T^{4} - 792 T^{5} + 16683 T^{6} - 212548 T^{7} + 639708 T^{8} + 6737848 T^{9} + 17936938 T^{10} - 60900622 T^{11} - 801992463 T^{12} - 60900622 p T^{13} + 17936938 p^{2} T^{14} + 6737848 p^{3} T^{15} + 639708 p^{4} T^{16} - 212548 p^{5} T^{17} + 16683 p^{6} T^{18} - 792 p^{7} T^{19} + 1002 p^{8} T^{20} + 130 p^{9} T^{21} - 34 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 18 T + 343 T^{2} + 3709 T^{3} + 40406 T^{4} + 309587 T^{5} + 2369267 T^{6} + 309587 p T^{7} + 40406 p^{2} T^{8} + 3709 p^{3} T^{9} + 343 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 + 16 T + 100 T^{2} - 3 T^{3} - 5957 T^{4} - 70768 T^{5} - 319774 T^{6} + 1357186 T^{7} + 28247303 T^{8} + 174575592 T^{9} + 229669305 T^{10} - 7818523957 T^{11} - 84912819061 T^{12} - 7818523957 p T^{13} + 229669305 p^{2} T^{14} + 174575592 p^{3} T^{15} + 28247303 p^{4} T^{16} + 1357186 p^{5} T^{17} - 319774 p^{6} T^{18} - 70768 p^{7} T^{19} - 5957 p^{8} T^{20} - 3 p^{9} T^{21} + 100 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 19 T + 116 T^{2} + 394 T^{3} - 9883 T^{4} + 26078 T^{5} + 480907 T^{6} - 4096060 T^{7} - 898122 T^{8} + 184584084 T^{9} - 811724803 T^{10} - 4795728867 T^{11} + 72032511241 T^{12} - 4795728867 p T^{13} - 811724803 p^{2} T^{14} + 184584084 p^{3} T^{15} - 898122 p^{4} T^{16} - 4096060 p^{5} T^{17} + 480907 p^{6} T^{18} + 26078 p^{7} T^{19} - 9883 p^{8} T^{20} + 394 p^{9} T^{21} + 116 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 46 T + 17 p T^{2} - 14790 T^{3} + 177614 T^{4} - 1817192 T^{5} + 15355586 T^{6} - 110059704 T^{7} + 724609191 T^{8} - 4321513829 T^{9} + 22893958688 T^{10} - 139528292361 T^{11} + 1061336056450 T^{12} - 139528292361 p T^{13} + 22893958688 p^{2} T^{14} - 4321513829 p^{3} T^{15} + 724609191 p^{4} T^{16} - 110059704 p^{5} T^{17} + 15355586 p^{6} T^{18} - 1817192 p^{7} T^{19} + 177614 p^{8} T^{20} - 14790 p^{9} T^{21} + 17 p^{11} T^{22} - 46 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 18 T + 143 T^{2} - 262 T^{3} - 2070 T^{4} + 48272 T^{5} - 674834 T^{6} + 5950496 T^{7} - 14470953 T^{8} - 59769648 T^{9} + 1431250730 T^{10} - 19727940960 T^{11} + 226152125887 T^{12} - 19727940960 p T^{13} + 1431250730 p^{2} T^{14} - 59769648 p^{3} T^{15} - 14470953 p^{4} T^{16} + 5950496 p^{5} T^{17} - 674834 p^{6} T^{18} + 48272 p^{7} T^{19} - 2070 p^{8} T^{20} - 262 p^{9} T^{21} + 143 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 22 T + 298 T^{2} + 3305 T^{3} + 30898 T^{4} + 261640 T^{5} + 2192889 T^{6} + 261640 p T^{7} + 30898 p^{2} T^{8} + 3305 p^{3} T^{9} + 298 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 6 T - 138 T^{2} - 150 T^{3} + 14830 T^{4} - 440 T^{5} - 527253 T^{6} + 1596758 T^{7} + 20994598 T^{8} + 218070950 T^{9} + 363856630 T^{10} - 3526845284 T^{11} + 80845701285 T^{12} - 3526845284 p T^{13} + 363856630 p^{2} T^{14} + 218070950 p^{3} T^{15} + 20994598 p^{4} T^{16} + 1596758 p^{5} T^{17} - 527253 p^{6} T^{18} - 440 p^{7} T^{19} + 14830 p^{8} T^{20} - 150 p^{9} T^{21} - 138 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 25 T + 251 T^{2} - 2032 T^{3} + 20889 T^{4} - 248774 T^{5} + 3257509 T^{6} - 31840018 T^{7} + 255450372 T^{8} - 2392308274 T^{9} + 21758412191 T^{10} - 207368759499 T^{11} + 1989322973227 T^{12} - 207368759499 p T^{13} + 21758412191 p^{2} T^{14} - 2392308274 p^{3} T^{15} + 255450372 p^{4} T^{16} - 31840018 p^{5} T^{17} + 3257509 p^{6} T^{18} - 248774 p^{7} T^{19} + 20889 p^{8} T^{20} - 2032 p^{9} T^{21} + 251 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 19 T + 191 T^{2} - 1563 T^{3} + 13533 T^{4} - 197906 T^{5} + 2500783 T^{6} - 25151932 T^{7} + 233332290 T^{8} - 2136669282 T^{9} + 20269148603 T^{10} - 2316670370 p T^{11} + 1604211446239 T^{12} - 2316670370 p^{2} T^{13} + 20269148603 p^{2} T^{14} - 2136669282 p^{3} T^{15} + 233332290 p^{4} T^{16} - 25151932 p^{5} T^{17} + 2500783 p^{6} T^{18} - 197906 p^{7} T^{19} + 13533 p^{8} T^{20} - 1563 p^{9} T^{21} + 191 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 69 T^{2} + 880 T^{3} + 6084 T^{4} - 45740 T^{5} - 19570 T^{6} + 5120870 T^{7} - 1828285 T^{8} - 334958985 T^{9} + 7758400176 T^{10} + 39051761105 T^{11} - 527014450634 T^{12} + 39051761105 p T^{13} + 7758400176 p^{2} T^{14} - 334958985 p^{3} T^{15} - 1828285 p^{4} T^{16} + 5120870 p^{5} T^{17} - 19570 p^{6} T^{18} - 45740 p^{7} T^{19} + 6084 p^{8} T^{20} + 880 p^{9} T^{21} - 69 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 25 T + 475 T^{2} + 6627 T^{3} + 81130 T^{4} + 837405 T^{5} + 8303699 T^{6} + 837405 p T^{7} + 81130 p^{2} T^{8} + 6627 p^{3} T^{9} + 475 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 31 T + 548 T^{2} + 9474 T^{3} + 162663 T^{4} + 2436400 T^{5} + 33137490 T^{6} + 429822937 T^{7} + 5352682356 T^{8} + 62584842515 T^{9} + 687098204439 T^{10} + 7284294048931 T^{11} + 74073360462526 T^{12} + 7284294048931 p T^{13} + 687098204439 p^{2} T^{14} + 62584842515 p^{3} T^{15} + 5352682356 p^{4} T^{16} + 429822937 p^{5} T^{17} + 33137490 p^{6} T^{18} + 2436400 p^{7} T^{19} + 162663 p^{8} T^{20} + 9474 p^{9} T^{21} + 548 p^{10} T^{22} + 31 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.64165716101120407943908824104, −3.59867020135047573339833956219, −3.55175359226655503761459740031, −3.50251028289401625836189004785, −3.38549619534007655656278034501, −3.21843196969202612812043662965, −3.05746723834901662415095793014, −3.05109022081433685013133410650, −2.84150416631865879019943857595, −2.76401357334458849190113290572, −2.48263114074135971625730581968, −2.44877552980077392737911227648, −2.40481881395112553175317911981, −2.22849091700687552250920150392, −2.12652503135943140189334162445, −1.91685447993736165627199061248, −1.58158974515677073654275041805, −1.50493210030109633307638404028, −1.47742410169180444556460482802, −1.42439928198283692083560730824, −1.22123034649820659415221622381, −1.20207517374725923348083962320, −0.927140316110651408393826867011, −0.70496052822937729456297940658, −0.11090017620865674285147845647, 0.11090017620865674285147845647, 0.70496052822937729456297940658, 0.927140316110651408393826867011, 1.20207517374725923348083962320, 1.22123034649820659415221622381, 1.42439928198283692083560730824, 1.47742410169180444556460482802, 1.50493210030109633307638404028, 1.58158974515677073654275041805, 1.91685447993736165627199061248, 2.12652503135943140189334162445, 2.22849091700687552250920150392, 2.40481881395112553175317911981, 2.44877552980077392737911227648, 2.48263114074135971625730581968, 2.76401357334458849190113290572, 2.84150416631865879019943857595, 3.05109022081433685013133410650, 3.05746723834901662415095793014, 3.21843196969202612812043662965, 3.38549619534007655656278034501, 3.50251028289401625836189004785, 3.55175359226655503761459740031, 3.59867020135047573339833956219, 3.64165716101120407943908824104
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.