Dirichlet series
| L(s) = 1 | − 6·2-s − 17·4-s − 50·7-s + 144·8-s + 269·9-s + 126·11-s + 300·14-s − 127·16-s − 1.61e3·18-s − 756·22-s − 756·23-s − 750·25-s + 850·28-s − 2.19e3·29-s − 1.05e3·32-s − 4.57e3·36-s + 5.56e3·37-s + 3.94e3·43-s − 2.14e3·44-s + 4.53e3·46-s − 3.14e3·49-s + 4.50e3·50-s + 1.17e4·53-s − 7.20e3·56-s + 1.31e4·58-s − 1.34e4·63-s + 1.17e4·64-s + ⋯ |
| L(s) = 1 | − 3/2·2-s − 1.06·4-s − 1.02·7-s + 9/4·8-s + 3.32·9-s + 1.04·11-s + 1.53·14-s − 0.496·16-s − 4.98·18-s − 1.56·22-s − 1.42·23-s − 6/5·25-s + 1.08·28-s − 2.60·29-s − 1.02·32-s − 3.52·36-s + 4.06·37-s + 2.13·43-s − 1.10·44-s + 2.14·46-s − 1.31·49-s + 9/5·50-s + 4.18·53-s − 2.29·56-s + 3.90·58-s − 3.38·63-s + 2.87·64-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.02977\times 10^{6}\) |
| Root analytic conductor: | \(1.90209\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{12} \cdot 7^{12} ,\ ( \ : [2]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.1793093218\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1793093218\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p^{3} T^{2} )^{6} \) |
| 7 | \( 1 + 50 T + 5648 T^{2} + 118950 T^{3} + 1457857 p T^{4} - 2750600 p^{2} T^{5} + 4017904 p^{4} T^{6} - 2750600 p^{6} T^{7} + 1457857 p^{9} T^{8} + 118950 p^{12} T^{9} + 5648 p^{16} T^{10} + 50 p^{20} T^{11} + p^{24} T^{12} \) | |
| good | 2 | \( ( 1 + 3 T + 11 p T^{2} + 9 p^{3} T^{3} + 113 p^{2} T^{4} + 489 p^{2} T^{5} + 899 p^{3} T^{6} + 489 p^{6} T^{7} + 113 p^{10} T^{8} + 9 p^{15} T^{9} + 11 p^{17} T^{10} + 3 p^{20} T^{11} + p^{24} T^{12} )^{2} \) |
| 3 | \( 1 - 269 T^{2} + 47947 T^{4} - 2295496 p T^{6} + 90502297 p^{2} T^{8} - 338177981 p^{5} T^{10} + 1086548878 p^{8} T^{12} - 338177981 p^{13} T^{14} + 90502297 p^{18} T^{16} - 2295496 p^{25} T^{18} + 47947 p^{32} T^{20} - 269 p^{40} T^{22} + p^{48} T^{24} \) | |
| 11 | \( ( 1 - 63 T + 27343 T^{2} - 2208684 T^{3} + 751431449 T^{4} - 48696366249 T^{5} + 10633083598006 T^{6} - 48696366249 p^{4} T^{7} + 751431449 p^{8} T^{8} - 2208684 p^{12} T^{9} + 27343 p^{16} T^{10} - 63 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 13 | \( 1 - 10173 p T^{2} + 770460159 p T^{4} - 570647485800148 T^{6} + 25761922467029808273 T^{8} - \)\(95\!\cdots\!83\)\( T^{10} + \)\(29\!\cdots\!18\)\( T^{12} - \)\(95\!\cdots\!83\)\( p^{8} T^{14} + 25761922467029808273 p^{16} T^{16} - 570647485800148 p^{24} T^{18} + 770460159 p^{33} T^{20} - 10173 p^{41} T^{22} + p^{48} T^{24} \) | |
| 17 | \( 1 - 556149 T^{2} + 151066985727 T^{4} - 27197046769603108 T^{6} + \)\(36\!\cdots\!13\)\( T^{8} - \)\(40\!\cdots\!03\)\( T^{10} + \)\(12\!\cdots\!62\)\( p^{2} T^{12} - \)\(40\!\cdots\!03\)\( p^{8} T^{14} + \)\(36\!\cdots\!13\)\( p^{16} T^{16} - 27197046769603108 p^{24} T^{18} + 151066985727 p^{32} T^{20} - 556149 p^{40} T^{22} + p^{48} T^{24} \) | |
| 19 | \( 1 - 894732 T^{2} + 359601659106 T^{4} - 83192357848521820 T^{6} + \)\(11\!\cdots\!95\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!64\)\( T^{12} - \)\(11\!\cdots\!92\)\( p^{8} T^{14} + \)\(11\!\cdots\!95\)\( p^{16} T^{16} - 83192357848521820 p^{24} T^{18} + 359601659106 p^{32} T^{20} - 894732 p^{40} T^{22} + p^{48} T^{24} \) | |
| 23 | \( ( 1 + 378 T + 1250032 T^{2} + 352644462 T^{3} + 716538259367 T^{4} + 6831761183352 p T^{5} + 248631042587552752 T^{6} + 6831761183352 p^{5} T^{7} + 716538259367 p^{8} T^{8} + 352644462 p^{12} T^{9} + 1250032 p^{16} T^{10} + 378 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 1095 T + 2793031 T^{2} + 84827028 p T^{3} + 3706824551861 T^{4} + 2851889617089573 T^{5} + 3215135226081063334 T^{6} + 2851889617089573 p^{4} T^{7} + 3706824551861 p^{8} T^{8} + 84827028 p^{13} T^{9} + 2793031 p^{16} T^{10} + 1095 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 31 | \( 1 - 6216432 T^{2} + 17972810788506 T^{4} - 33582383115032150320 T^{6} + \)\(48\!\cdots\!95\)\( T^{8} - \)\(57\!\cdots\!92\)\( T^{10} + \)\(58\!\cdots\!64\)\( T^{12} - \)\(57\!\cdots\!92\)\( p^{8} T^{14} + \)\(48\!\cdots\!95\)\( p^{16} T^{16} - 33582383115032150320 p^{24} T^{18} + 17972810788506 p^{32} T^{20} - 6216432 p^{40} T^{22} + p^{48} T^{24} \) | |
| 37 | \( ( 1 - 2782 T + 9009842 T^{2} - 19591734318 T^{3} + 38548074098527 T^{4} - 62563245467301524 T^{5} + 93912362156791592092 T^{6} - 62563245467301524 p^{4} T^{7} + 38548074098527 p^{8} T^{8} - 19591734318 p^{12} T^{9} + 9009842 p^{16} T^{10} - 2782 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 41 | \( 1 - 14414952 T^{2} + 108169094529786 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!95\)\( T^{8} - \)\(98\!\cdots\!92\)\( T^{10} + \)\(30\!\cdots\!04\)\( T^{12} - \)\(98\!\cdots\!92\)\( p^{8} T^{14} + \)\(26\!\cdots\!95\)\( p^{16} T^{16} - \)\(59\!\cdots\!20\)\( p^{24} T^{18} + 108169094529786 p^{32} T^{20} - 14414952 p^{40} T^{22} + p^{48} T^{24} \) | |
| 43 | \( ( 1 - 1972 T + 13184912 T^{2} - 22921064628 T^{3} + 95179380615367 T^{4} - 134435669219868824 T^{5} + \)\(40\!\cdots\!12\)\( T^{6} - 134435669219868824 p^{4} T^{7} + 95179380615367 p^{8} T^{8} - 22921064628 p^{12} T^{9} + 13184912 p^{16} T^{10} - 1972 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 47 | \( 1 - 29944929 T^{2} + 489542731138407 T^{4} - \)\(55\!\cdots\!68\)\( T^{6} + \)\(47\!\cdots\!53\)\( T^{8} - \)\(32\!\cdots\!43\)\( T^{10} + \)\(17\!\cdots\!78\)\( T^{12} - \)\(32\!\cdots\!43\)\( p^{8} T^{14} + \)\(47\!\cdots\!53\)\( p^{16} T^{16} - \)\(55\!\cdots\!68\)\( p^{24} T^{18} + 489542731138407 p^{32} T^{20} - 29944929 p^{40} T^{22} + p^{48} T^{24} \) | |
| 53 | \( ( 1 - 5880 T + 58645978 T^{2} - 238394366040 T^{3} + 1287971516424239 T^{4} - 3810587664196645680 T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - 3810587664196645680 p^{4} T^{7} + 1287971516424239 p^{8} T^{8} - 238394366040 p^{12} T^{9} + 58645978 p^{16} T^{10} - 5880 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 59 | \( 1 - 65287452 T^{2} + 1781241115993986 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!95\)\( T^{8} - \)\(58\!\cdots\!92\)\( T^{10} + \)\(83\!\cdots\!04\)\( T^{12} - \)\(58\!\cdots\!92\)\( p^{8} T^{14} + \)\(37\!\cdots\!95\)\( p^{16} T^{16} - \)\(28\!\cdots\!20\)\( p^{24} T^{18} + 1781241115993986 p^{32} T^{20} - 65287452 p^{40} T^{22} + p^{48} T^{24} \) | |
| 61 | \( 1 - 42658512 T^{2} + 1040808855512346 T^{4} - \)\(21\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!95\)\( T^{8} - \)\(64\!\cdots\!92\)\( T^{10} + \)\(94\!\cdots\!24\)\( T^{12} - \)\(64\!\cdots\!92\)\( p^{8} T^{14} + \)\(39\!\cdots\!95\)\( p^{16} T^{16} - \)\(21\!\cdots\!20\)\( p^{24} T^{18} + 1040808855512346 p^{32} T^{20} - 42658512 p^{40} T^{22} + p^{48} T^{24} \) | |
| 67 | \( ( 1 + 12048 T + 89957712 T^{2} + 425874915752 T^{3} + 2163775124518407 T^{4} + 11220458774359181256 T^{5} + \)\(58\!\cdots\!32\)\( T^{6} + 11220458774359181256 p^{4} T^{7} + 2163775124518407 p^{8} T^{8} + 425874915752 p^{12} T^{9} + 89957712 p^{16} T^{10} + 12048 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 2832 T + 99343678 T^{2} + 227069165136 T^{3} + 4836254642980799 T^{4} + 8840855916674095776 T^{5} + \)\(15\!\cdots\!56\)\( T^{6} + 8840855916674095776 p^{4} T^{7} + 4836254642980799 p^{8} T^{8} + 227069165136 p^{12} T^{9} + 99343678 p^{16} T^{10} + 2832 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 73 | \( 1 - 201646824 T^{2} + 18852404589378042 T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(47\!\cdots\!23\)\( T^{8} - \)\(16\!\cdots\!88\)\( T^{10} + \)\(50\!\cdots\!68\)\( T^{12} - \)\(16\!\cdots\!88\)\( p^{8} T^{14} + \)\(47\!\cdots\!23\)\( p^{16} T^{16} - \)\(11\!\cdots\!08\)\( p^{24} T^{18} + 18852404589378042 p^{32} T^{20} - 201646824 p^{40} T^{22} + p^{48} T^{24} \) | |
| 79 | \( ( 1 + 795 T + 141722871 T^{2} - 71653153828 T^{3} + 9308937946787721 T^{4} - 13513763479751036667 T^{5} + \)\(41\!\cdots\!74\)\( T^{6} - 13513763479751036667 p^{4} T^{7} + 9308937946787721 p^{8} T^{8} - 71653153828 p^{12} T^{9} + 141722871 p^{16} T^{10} + 795 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 83 | \( 1 - 392302704 T^{2} + 74776635513055182 T^{4} - \)\(92\!\cdots\!48\)\( T^{6} + \)\(81\!\cdots\!03\)\( T^{8} - \)\(55\!\cdots\!08\)\( T^{10} + \)\(29\!\cdots\!28\)\( T^{12} - \)\(55\!\cdots\!08\)\( p^{8} T^{14} + \)\(81\!\cdots\!03\)\( p^{16} T^{16} - \)\(92\!\cdots\!48\)\( p^{24} T^{18} + 74776635513055182 p^{32} T^{20} - 392302704 p^{40} T^{22} + p^{48} T^{24} \) | |
| 89 | \( 1 - 380275212 T^{2} + 70690585417318146 T^{4} - \)\(87\!\cdots\!20\)\( T^{6} + \)\(81\!\cdots\!95\)\( T^{8} - \)\(63\!\cdots\!92\)\( T^{10} + \)\(42\!\cdots\!24\)\( T^{12} - \)\(63\!\cdots\!92\)\( p^{8} T^{14} + \)\(81\!\cdots\!95\)\( p^{16} T^{16} - \)\(87\!\cdots\!20\)\( p^{24} T^{18} + 70690585417318146 p^{32} T^{20} - 380275212 p^{40} T^{22} + p^{48} T^{24} \) | |
| 97 | \( 1 - 308934069 T^{2} + 30796677588355647 T^{4} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(30\!\cdots\!73\)\( T^{8} + \)\(30\!\cdots\!61\)\( p T^{10} - \)\(86\!\cdots\!42\)\( T^{12} + \)\(30\!\cdots\!61\)\( p^{9} T^{14} + \)\(30\!\cdots\!73\)\( p^{16} T^{16} - \)\(10\!\cdots\!88\)\( p^{24} T^{18} + 30796677588355647 p^{32} T^{20} - 308934069 p^{40} T^{22} + p^{48} T^{24} \) | |
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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
−5.78131771745191043063098567108, −5.69459200109345377387451812327, −5.17876984572570852307570623610, −4.83013286727816391897178659131, −4.71984786119418643442986367946, −4.60305032625780374665068803449, −4.50940102440545341903793585334, −4.44024724820835891657827730119, −4.23499045538410648653437696190, −4.12752483442919311343796299201, −3.89109306753690178997110248438, −3.79427199849119524054647088234, −3.55341040099869119571570346788, −3.51599073321578748746783719278, −3.19660428043204240013767427160, −2.78168034146106034682653541541, −2.42810823308654331708997695685, −2.35710335076475836682733424262, −1.94965141984434777110252163154, −1.78068438801368052241908468486, −1.44563650547827161501824083352, −1.17411792144543628717919002084, −0.798391254824136071476031379032, −0.49832222167753604881300073646, −0.12712121021265677767703431417, 0.12712121021265677767703431417, 0.49832222167753604881300073646, 0.798391254824136071476031379032, 1.17411792144543628717919002084, 1.44563650547827161501824083352, 1.78068438801368052241908468486, 1.94965141984434777110252163154, 2.35710335076475836682733424262, 2.42810823308654331708997695685, 2.78168034146106034682653541541, 3.19660428043204240013767427160, 3.51599073321578748746783719278, 3.55341040099869119571570346788, 3.79427199849119524054647088234, 3.89109306753690178997110248438, 4.12752483442919311343796299201, 4.23499045538410648653437696190, 4.44024724820835891657827730119, 4.50940102440545341903793585334, 4.60305032625780374665068803449, 4.71984786119418643442986367946, 4.83013286727816391897178659131, 5.17876984572570852307570623610, 5.69459200109345377387451812327, 5.78131771745191043063098567108
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.