Dirichlet series
| L(s) = 1 | + 50·4-s + 180·11-s − 216·13-s + 1.15e3·16-s − 678·17-s − 1.56e3·23-s + 3.66e3·25-s + 5.71e3·31-s − 4.87e3·41-s − 1.10e3·43-s + 9.00e3·44-s + 5.52e3·47-s + 1.01e4·49-s − 1.08e4·52-s − 1.21e3·53-s − 1.40e4·59-s + 1.61e4·64-s − 1.08e3·67-s − 3.39e4·68-s + 2.43e4·79-s + 7.03e3·83-s − 7.83e4·92-s − 5.84e3·97-s + 1.83e5·100-s + 2.52e4·101-s − 2.79e4·103-s + 4.10e3·107-s + ⋯ |
| L(s) = 1 | + 25/8·4-s + 1.48·11-s − 1.27·13-s + 4.49·16-s − 2.34·17-s − 2.96·23-s + 5.86·25-s + 5.94·31-s − 2.90·41-s − 0.599·43-s + 4.64·44-s + 2.50·47-s + 4.22·49-s − 3.99·52-s − 0.431·53-s − 4.02·59-s + 3.94·64-s − 0.242·67-s − 7.33·68-s + 3.89·79-s + 1.02·83-s − 9.25·92-s − 0.620·97-s + 18.3·100-s + 2.47·101-s − 2.63·103-s + 0.358·107-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 43^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 43^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.67982\times 10^{19}\) |
| Root analytic conductor: | \(6.32488\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 43^{12} ,\ ( \ : [2]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(25.71671091\) |
| \(L(\frac12)\) | \(\approx\) | \(25.71671091\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 1108 T + 44122 p T^{2} - 2531708 p^{2} T^{3} + 8565509 p^{3} T^{4} - 247619960 p^{4} T^{5} + 10220444708 p^{6} T^{6} - 247619960 p^{8} T^{7} + 8565509 p^{11} T^{8} - 2531708 p^{14} T^{9} + 44122 p^{17} T^{10} + 1108 p^{20} T^{11} + p^{24} T^{12} \) | |
| good | 2 | \( 1 - 25 p T^{2} + 1349 T^{4} - 3257 p^{3} T^{6} + 3619 p^{7} T^{8} - 136749 p^{6} T^{10} + 9432207 p^{4} T^{12} - 136749 p^{14} T^{14} + 3619 p^{23} T^{16} - 3257 p^{27} T^{18} + 1349 p^{32} T^{20} - 25 p^{41} T^{22} + p^{48} T^{24} \) |
| 5 | \( 1 - 3663 T^{2} + 1302203 p T^{4} - 1473032153 p T^{6} + 5973224140031 T^{8} - 3902236317085676 T^{10} + 2399998678381934162 T^{12} - 3902236317085676 p^{8} T^{14} + 5973224140031 p^{16} T^{16} - 1473032153 p^{25} T^{18} + 1302203 p^{33} T^{20} - 3663 p^{40} T^{22} + p^{48} T^{24} \) | |
| 7 | \( 1 - 10134 T^{2} + 60060006 T^{4} - 249160434566 T^{6} + 812915964019071 T^{8} - 2246049549704995236 T^{10} + \)\(56\!\cdots\!48\)\( T^{12} - 2246049549704995236 p^{8} T^{14} + 812915964019071 p^{16} T^{16} - 249160434566 p^{24} T^{18} + 60060006 p^{32} T^{20} - 10134 p^{40} T^{22} + p^{48} T^{24} \) | |
| 11 | \( ( 1 - 90 T + 69626 T^{2} - 4275732 T^{3} + 2084651350 T^{4} - 8260523922 p T^{5} + 309663152918 p^{2} T^{6} - 8260523922 p^{5} T^{7} + 2084651350 p^{8} T^{8} - 4275732 p^{12} T^{9} + 69626 p^{16} T^{10} - 90 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 13 | \( ( 1 + 108 T + 6010 p T^{2} + 14124934 T^{3} + 3639591614 T^{4} + 705010114760 T^{5} + 121575713991590 T^{6} + 705010114760 p^{4} T^{7} + 3639591614 p^{8} T^{8} + 14124934 p^{12} T^{9} + 6010 p^{17} T^{10} + 108 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 17 | \( ( 1 + 339 T + 345843 T^{2} + 6101049 p T^{3} + 57770879546 T^{4} + 15179725016055 T^{5} + 6016834256618483 T^{6} + 15179725016055 p^{4} T^{7} + 57770879546 p^{8} T^{8} + 6101049 p^{13} T^{9} + 345843 p^{16} T^{10} + 339 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 19 | \( 1 - 1175657 T^{2} + 671010023409 T^{4} - 244979577528319381 T^{6} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!98\)\( T^{12} - \)\(12\!\cdots\!10\)\( p^{8} T^{14} + \)\(63\!\cdots\!63\)\( p^{16} T^{16} - 244979577528319381 p^{24} T^{18} + 671010023409 p^{32} T^{20} - 1175657 p^{40} T^{22} + p^{48} T^{24} \) | |
| 23 | \( ( 1 + 783 T + 1163077 T^{2} + 843582057 T^{3} + 713091466102 T^{4} + 397049506101027 T^{5} + 259574077696956889 T^{6} + 397049506101027 p^{4} T^{7} + 713091466102 p^{8} T^{8} + 843582057 p^{12} T^{9} + 1163077 p^{16} T^{10} + 783 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 29 | \( 1 - 5231219 T^{2} + 468703896099 p T^{4} - 23367533349246758977 T^{6} + \)\(29\!\cdots\!35\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!82\)\( T^{12} - \)\(29\!\cdots\!88\)\( p^{8} T^{14} + \)\(29\!\cdots\!35\)\( p^{16} T^{16} - 23367533349246758977 p^{24} T^{18} + 468703896099 p^{33} T^{20} - 5231219 p^{40} T^{22} + p^{48} T^{24} \) | |
| 31 | \( ( 1 - 2855 T + 6518969 T^{2} - 8600011177 T^{3} + 10228398490358 T^{4} - 8682474860253963 T^{5} + 8850492205571119245 T^{6} - 8682474860253963 p^{4} T^{7} + 10228398490358 p^{8} T^{8} - 8600011177 p^{12} T^{9} + 6518969 p^{16} T^{10} - 2855 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 37 | \( 1 - 6978249 T^{2} + 25643197308697 T^{4} - 70260487375765593853 T^{6} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(39\!\cdots\!70\)\( T^{10} + \)\(80\!\cdots\!70\)\( T^{12} - \)\(39\!\cdots\!70\)\( p^{8} T^{14} + \)\(17\!\cdots\!59\)\( p^{16} T^{16} - 70260487375765593853 p^{24} T^{18} + 25643197308697 p^{32} T^{20} - 6978249 p^{40} T^{22} + p^{48} T^{24} \) | |
| 41 | \( ( 1 + 2439 T + 13822603 T^{2} + 19943096547 T^{3} + 69051390319336 T^{4} + 65595635499999207 T^{5} + \)\(21\!\cdots\!21\)\( T^{6} + 65595635499999207 p^{4} T^{7} + 69051390319336 p^{8} T^{8} + 19943096547 p^{12} T^{9} + 13822603 p^{16} T^{10} + 2439 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 47 | \( ( 1 - 2763 T + 21754149 T^{2} - 55851382377 T^{3} + 234662825346711 T^{4} - 481604898621884868 T^{5} + \)\(14\!\cdots\!34\)\( T^{6} - 481604898621884868 p^{4} T^{7} + 234662825346711 p^{8} T^{8} - 55851382377 p^{12} T^{9} + 21754149 p^{16} T^{10} - 2763 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 606 T + 25718024 T^{2} + 24674633094 T^{3} + 346581142619392 T^{4} + 336085601563787574 T^{5} + \)\(32\!\cdots\!82\)\( T^{6} + 336085601563787574 p^{4} T^{7} + 346581142619392 p^{8} T^{8} + 24674633094 p^{12} T^{9} + 25718024 p^{16} T^{10} + 606 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 59 | \( ( 1 + 7008 T + 80137258 T^{2} + 394451068512 T^{3} + 2532059909380847 T^{4} + 9247600606403540352 T^{5} + \)\(41\!\cdots\!80\)\( T^{6} + 9247600606403540352 p^{4} T^{7} + 2532059909380847 p^{8} T^{8} + 394451068512 p^{12} T^{9} + 80137258 p^{16} T^{10} + 7008 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 61 | \( 1 - 32665486 T^{2} + 1118892201421590 T^{4} - \)\(24\!\cdots\!14\)\( T^{6} + \)\(52\!\cdots\!07\)\( T^{8} - \)\(83\!\cdots\!88\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{12} - \)\(83\!\cdots\!88\)\( p^{8} T^{14} + \)\(52\!\cdots\!07\)\( p^{16} T^{16} - \)\(24\!\cdots\!14\)\( p^{24} T^{18} + 1118892201421590 p^{32} T^{20} - 32665486 p^{40} T^{22} + p^{48} T^{24} \) | |
| 67 | \( ( 1 + 544 T + 66684038 T^{2} + 18315837698 T^{3} + 2452178722840730 T^{4} + 587634295199545788 T^{5} + \)\(60\!\cdots\!70\)\( T^{6} + 587634295199545788 p^{4} T^{7} + 2452178722840730 p^{8} T^{8} + 18315837698 p^{12} T^{9} + 66684038 p^{16} T^{10} + 544 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 71 | \( 1 - 101562020 T^{2} + 5784818387848562 T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(47\!\cdots\!67\)\( T^{8} - \)\(75\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(75\!\cdots\!84\)\( p^{8} T^{14} + \)\(47\!\cdots\!67\)\( p^{16} T^{16} - \)\(20\!\cdots\!64\)\( p^{24} T^{18} + 5784818387848562 p^{32} T^{20} - 101562020 p^{40} T^{22} + p^{48} T^{24} \) | |
| 73 | \( 1 - 193491698 T^{2} + 19000747654703030 T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!56\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(23\!\cdots\!56\)\( p^{8} T^{14} + \)\(60\!\cdots\!75\)\( p^{16} T^{16} - \)\(12\!\cdots\!18\)\( p^{24} T^{18} + 19000747654703030 p^{32} T^{20} - 193491698 p^{40} T^{22} + p^{48} T^{24} \) | |
| 79 | \( ( 1 - 12151 T + 155715789 T^{2} - 10136983391 p T^{3} + 4128429862975343 T^{4} + 5685482156720652576 T^{5} - \)\(30\!\cdots\!42\)\( T^{6} + 5685482156720652576 p^{4} T^{7} + 4128429862975343 p^{8} T^{8} - 10136983391 p^{13} T^{9} + 155715789 p^{16} T^{10} - 12151 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 83 | \( ( 1 - 3516 T + 117162268 T^{2} - 863865543048 T^{3} + 7914887456622244 T^{4} - 75598926847021341108 T^{5} + \)\(40\!\cdots\!22\)\( T^{6} - 75598926847021341108 p^{4} T^{7} + 7914887456622244 p^{8} T^{8} - 863865543048 p^{12} T^{9} + 117162268 p^{16} T^{10} - 3516 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 89 | \( 1 - 474457834 T^{2} + 1254552724574358 p T^{4} - \)\(17\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(17\!\cdots\!92\)\( p^{8} T^{14} + \)\(19\!\cdots\!11\)\( p^{16} T^{16} - \)\(17\!\cdots\!66\)\( p^{24} T^{18} + 1254552724574358 p^{33} T^{20} - 474457834 p^{40} T^{22} + p^{48} T^{24} \) | |
| 97 | \( ( 1 + 2921 T + 384124299 T^{2} + 837647682043 T^{3} + 69794294829141482 T^{4} + \)\(11\!\cdots\!33\)\( T^{5} + \)\(76\!\cdots\!95\)\( T^{6} + \)\(11\!\cdots\!33\)\( p^{4} T^{7} + 69794294829141482 p^{8} T^{8} + 837647682043 p^{12} T^{9} + 384124299 p^{16} T^{10} + 2921 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
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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
−3.03876795644483399811647391654, −2.73339645624159899700040067304, −2.72777023059354519290006054354, −2.68779821358731099129893254950, −2.60673009032133549999537775833, −2.49880309908621744402782337666, −2.43658549041062159510528519134, −2.42998077974691466028076408877, −2.25136198510256681262273227916, −2.17195038929045578339873809234, −1.93026701749086385180003894651, −1.82490054733683233381669070392, −1.62943602244390280631334177963, −1.58764373160799065391917418085, −1.56767362324568181784524254789, −1.44132666978241931236278331825, −1.08321772904586050219929221390, −0.985852064485153469616146880462, −0.880696327777674589899255986533, −0.868949316201025504178161130801, −0.834167670446944956224127264603, −0.69217930902843848953149460929, −0.36146952907815603256000221237, −0.20183426346292785146182295993, −0.13238644615401239651160241918, 0.13238644615401239651160241918, 0.20183426346292785146182295993, 0.36146952907815603256000221237, 0.69217930902843848953149460929, 0.834167670446944956224127264603, 0.868949316201025504178161130801, 0.880696327777674589899255986533, 0.985852064485153469616146880462, 1.08321772904586050219929221390, 1.44132666978241931236278331825, 1.56767362324568181784524254789, 1.58764373160799065391917418085, 1.62943602244390280631334177963, 1.82490054733683233381669070392, 1.93026701749086385180003894651, 2.17195038929045578339873809234, 2.25136198510256681262273227916, 2.42998077974691466028076408877, 2.43658549041062159510528519134, 2.49880309908621744402782337666, 2.60673009032133549999537775833, 2.68779821358731099129893254950, 2.72777023059354519290006054354, 2.73339645624159899700040067304, 3.03876795644483399811647391654
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.