Dirichlet series
| L(s) = 1 | − 1.25e7·2-s + 5.88e9·3-s + 9.23e13·4-s + 1.10e15·5-s − 7.40e16·6-s − 3.35e18·7-s − 5.16e20·8-s − 5.07e20·9-s − 1.38e22·10-s + 1.23e22·11-s + 5.43e23·12-s + 2.67e24·13-s + 4.21e25·14-s + 6.48e24·15-s + 2.43e27·16-s − 1.51e26·17-s + 6.38e27·18-s + 2.92e27·19-s + 1.01e29·20-s − 1.97e28·21-s − 1.55e29·22-s − 1.38e29·23-s − 3.04e30·24-s − 3.07e30·25-s − 3.36e31·26-s − 9.77e30·27-s − 3.09e32·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 0.324·3-s + 21/2·4-s + 1.03·5-s − 1.37·6-s − 2.26·7-s − 19.7·8-s − 1.54·9-s − 4.38·10-s + 0.503·11-s + 3.41·12-s + 3.00·13-s + 9.62·14-s + 0.335·15-s + 63/2·16-s − 0.531·17-s + 6.55·18-s + 0.939·19-s + 10.8·20-s − 0.737·21-s − 2.13·22-s − 0.730·23-s − 6.43·24-s − 2.70·25-s − 12.7·26-s − 1.64·27-s − 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7529536 ^{s/2} \, \Gamma_{\C}(s+43/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(7529536\) = \(2^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.94240\times 10^{13}\) |
| Root analytic conductor: | \(12.8044\) |
| Motivic weight: | \(43\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 7529536,\ (\ :[43/2]^{6}),\ 1)\) |
Particular Values
| \(L(22)\) | \(\approx\) | \(0.7682999163\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7682999163\) |
| \(L(\frac{45}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{21} T )^{6} \) |
| 7 | \( ( 1 + p^{21} T )^{6} \) | |
| good | 3 | \( 1 - 5888073464 T + 6691389857163386618 p^{4} T^{2} + \)\(49\!\cdots\!88\)\( p^{6} T^{3} + \)\(14\!\cdots\!49\)\( p^{13} T^{4} + \)\(80\!\cdots\!72\)\( p^{22} T^{5} + \)\(38\!\cdots\!04\)\( p^{32} T^{6} + \)\(80\!\cdots\!72\)\( p^{65} T^{7} + \)\(14\!\cdots\!49\)\( p^{99} T^{8} + \)\(49\!\cdots\!88\)\( p^{135} T^{9} + 6691389857163386618 p^{176} T^{10} - 5888073464 p^{215} T^{11} + p^{258} T^{12} \) |
| 5 | \( 1 - 1101555913947636 T + \)\(17\!\cdots\!46\)\( p^{2} T^{2} - \)\(23\!\cdots\!04\)\( p^{2} T^{3} + \)\(30\!\cdots\!79\)\( p^{5} T^{4} - \)\(12\!\cdots\!92\)\( p^{10} T^{5} + \)\(87\!\cdots\!96\)\( p^{16} T^{6} - \)\(12\!\cdots\!92\)\( p^{53} T^{7} + \)\(30\!\cdots\!79\)\( p^{91} T^{8} - \)\(23\!\cdots\!04\)\( p^{131} T^{9} + \)\(17\!\cdots\!46\)\( p^{174} T^{10} - 1101555913947636 p^{215} T^{11} + p^{258} T^{12} \) | |
| 11 | \( 1 - \)\(12\!\cdots\!56\)\( T + \)\(10\!\cdots\!62\)\( p^{2} T^{2} + \)\(36\!\cdots\!68\)\( p^{2} T^{3} + \)\(65\!\cdots\!93\)\( p^{5} T^{4} - \)\(85\!\cdots\!92\)\( p^{8} T^{5} + \)\(33\!\cdots\!28\)\( p^{11} T^{6} - \)\(85\!\cdots\!92\)\( p^{51} T^{7} + \)\(65\!\cdots\!93\)\( p^{91} T^{8} + \)\(36\!\cdots\!68\)\( p^{131} T^{9} + \)\(10\!\cdots\!62\)\( p^{174} T^{10} - \)\(12\!\cdots\!56\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 13 | \( 1 - \)\(26\!\cdots\!52\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(24\!\cdots\!32\)\( p T^{3} + \)\(13\!\cdots\!47\)\( p^{3} T^{4} - \)\(96\!\cdots\!32\)\( p^{5} T^{5} + \)\(49\!\cdots\!92\)\( p^{8} T^{6} - \)\(96\!\cdots\!32\)\( p^{48} T^{7} + \)\(13\!\cdots\!47\)\( p^{89} T^{8} - \)\(24\!\cdots\!32\)\( p^{130} T^{9} + \)\(38\!\cdots\!78\)\( p^{172} T^{10} - \)\(26\!\cdots\!52\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 17 | \( 1 + \)\(15\!\cdots\!36\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(34\!\cdots\!80\)\( p T^{3} + \)\(58\!\cdots\!35\)\( p^{3} T^{4} + \)\(49\!\cdots\!68\)\( p^{5} T^{5} + \)\(73\!\cdots\!52\)\( p^{7} T^{6} + \)\(49\!\cdots\!68\)\( p^{48} T^{7} + \)\(58\!\cdots\!35\)\( p^{89} T^{8} + \)\(34\!\cdots\!80\)\( p^{130} T^{9} + \)\(21\!\cdots\!18\)\( p^{172} T^{10} + \)\(15\!\cdots\!36\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 19 | \( 1 - \)\(29\!\cdots\!88\)\( T + \)\(20\!\cdots\!62\)\( p T^{2} - \)\(29\!\cdots\!52\)\( p^{2} T^{3} + \)\(57\!\cdots\!75\)\( p^{4} T^{4} - \)\(36\!\cdots\!44\)\( p^{6} T^{5} + \)\(53\!\cdots\!92\)\( p^{8} T^{6} - \)\(36\!\cdots\!44\)\( p^{49} T^{7} + \)\(57\!\cdots\!75\)\( p^{90} T^{8} - \)\(29\!\cdots\!52\)\( p^{131} T^{9} + \)\(20\!\cdots\!62\)\( p^{173} T^{10} - \)\(29\!\cdots\!88\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 23 | \( 1 + \)\(13\!\cdots\!44\)\( T + \)\(11\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!29\)\( p T^{4} + \)\(10\!\cdots\!60\)\( p^{2} T^{5} + \)\(24\!\cdots\!28\)\( p^{3} T^{6} + \)\(10\!\cdots\!60\)\( p^{45} T^{7} + \)\(30\!\cdots\!29\)\( p^{87} T^{8} + \)\(10\!\cdots\!20\)\( p^{129} T^{9} + \)\(11\!\cdots\!26\)\( p^{172} T^{10} + \)\(13\!\cdots\!44\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 29 | \( 1 - \)\(15\!\cdots\!04\)\( T + \)\(13\!\cdots\!70\)\( T^{2} - \)\(16\!\cdots\!56\)\( p T^{3} + \)\(18\!\cdots\!67\)\( p^{2} T^{4} - \)\(14\!\cdots\!12\)\( p^{3} T^{5} + \)\(24\!\cdots\!44\)\( p^{4} T^{6} - \)\(14\!\cdots\!12\)\( p^{46} T^{7} + \)\(18\!\cdots\!67\)\( p^{88} T^{8} - \)\(16\!\cdots\!56\)\( p^{130} T^{9} + \)\(13\!\cdots\!70\)\( p^{172} T^{10} - \)\(15\!\cdots\!04\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 31 | \( 1 - \)\(29\!\cdots\!28\)\( T + \)\(20\!\cdots\!62\)\( p T^{2} - \)\(97\!\cdots\!24\)\( p^{2} T^{3} + \)\(47\!\cdots\!85\)\( p^{3} T^{4} - \)\(20\!\cdots\!52\)\( p^{4} T^{5} + \)\(83\!\cdots\!52\)\( p^{5} T^{6} - \)\(20\!\cdots\!52\)\( p^{47} T^{7} + \)\(47\!\cdots\!85\)\( p^{89} T^{8} - \)\(97\!\cdots\!24\)\( p^{131} T^{9} + \)\(20\!\cdots\!62\)\( p^{173} T^{10} - \)\(29\!\cdots\!28\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 37 | \( 1 - \)\(13\!\cdots\!64\)\( T + \)\(26\!\cdots\!34\)\( p T^{2} - \)\(33\!\cdots\!60\)\( p^{2} T^{3} + \)\(18\!\cdots\!35\)\( p^{3} T^{4} + \)\(32\!\cdots\!16\)\( p^{4} T^{5} - \)\(77\!\cdots\!72\)\( p^{5} T^{6} + \)\(32\!\cdots\!16\)\( p^{47} T^{7} + \)\(18\!\cdots\!35\)\( p^{89} T^{8} - \)\(33\!\cdots\!60\)\( p^{131} T^{9} + \)\(26\!\cdots\!34\)\( p^{173} T^{10} - \)\(13\!\cdots\!64\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 41 | \( 1 - \)\(95\!\cdots\!48\)\( p T + \)\(82\!\cdots\!42\)\( p^{2} T^{2} - \)\(16\!\cdots\!44\)\( p^{3} T^{3} + \)\(16\!\cdots\!35\)\( p^{4} T^{4} - \)\(12\!\cdots\!12\)\( p^{5} T^{5} + \)\(16\!\cdots\!52\)\( p^{6} T^{6} - \)\(12\!\cdots\!12\)\( p^{48} T^{7} + \)\(16\!\cdots\!35\)\( p^{90} T^{8} - \)\(16\!\cdots\!44\)\( p^{132} T^{9} + \)\(82\!\cdots\!42\)\( p^{174} T^{10} - \)\(95\!\cdots\!48\)\( p^{216} T^{11} + p^{258} T^{12} \) | |
| 43 | \( 1 - \)\(43\!\cdots\!80\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(41\!\cdots\!91\)\( T^{4} - \)\(58\!\cdots\!08\)\( T^{5} + \)\(76\!\cdots\!96\)\( T^{6} - \)\(58\!\cdots\!08\)\( p^{43} T^{7} + \)\(41\!\cdots\!91\)\( p^{86} T^{8} - \)\(25\!\cdots\!36\)\( p^{129} T^{9} + \)\(12\!\cdots\!46\)\( p^{172} T^{10} - \)\(43\!\cdots\!80\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 47 | \( 1 + \)\(78\!\cdots\!04\)\( T + \)\(24\!\cdots\!82\)\( T^{2} + \)\(14\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!59\)\( T^{4} + \)\(19\!\cdots\!12\)\( T^{5} + \)\(32\!\cdots\!28\)\( T^{6} + \)\(19\!\cdots\!12\)\( p^{43} T^{7} + \)\(32\!\cdots\!59\)\( p^{86} T^{8} + \)\(14\!\cdots\!28\)\( p^{129} T^{9} + \)\(24\!\cdots\!82\)\( p^{172} T^{10} + \)\(78\!\cdots\!04\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 53 | \( 1 + \)\(51\!\cdots\!20\)\( T + \)\(49\!\cdots\!46\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!11\)\( T^{4} + \)\(42\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!76\)\( T^{6} + \)\(42\!\cdots\!52\)\( p^{43} T^{7} + \)\(12\!\cdots\!11\)\( p^{86} T^{8} + \)\(19\!\cdots\!84\)\( p^{129} T^{9} + \)\(49\!\cdots\!46\)\( p^{172} T^{10} + \)\(51\!\cdots\!20\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 59 | \( 1 + \)\(22\!\cdots\!68\)\( T + \)\(77\!\cdots\!58\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!55\)\( T^{4} + \)\(50\!\cdots\!56\)\( p T^{5} + \)\(45\!\cdots\!12\)\( T^{6} + \)\(50\!\cdots\!56\)\( p^{44} T^{7} + \)\(24\!\cdots\!55\)\( p^{86} T^{8} + \)\(11\!\cdots\!12\)\( p^{129} T^{9} + \)\(77\!\cdots\!58\)\( p^{172} T^{10} + \)\(22\!\cdots\!68\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 61 | \( 1 + \)\(28\!\cdots\!36\)\( T + \)\(89\!\cdots\!30\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!71\)\( T^{4} + \)\(51\!\cdots\!40\)\( T^{5} + \)\(72\!\cdots\!24\)\( T^{6} + \)\(51\!\cdots\!40\)\( p^{43} T^{7} + \)\(13\!\cdots\!71\)\( p^{86} T^{8} - \)\(13\!\cdots\!60\)\( p^{129} T^{9} + \)\(89\!\cdots\!30\)\( p^{172} T^{10} + \)\(28\!\cdots\!36\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 67 | \( 1 - \)\(34\!\cdots\!84\)\( T + \)\(13\!\cdots\!18\)\( T^{2} + \)\(56\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!55\)\( T^{4} + \)\(22\!\cdots\!76\)\( T^{5} + \)\(32\!\cdots\!76\)\( T^{6} + \)\(22\!\cdots\!76\)\( p^{43} T^{7} + \)\(83\!\cdots\!55\)\( p^{86} T^{8} + \)\(56\!\cdots\!60\)\( p^{129} T^{9} + \)\(13\!\cdots\!18\)\( p^{172} T^{10} - \)\(34\!\cdots\!84\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 71 | \( 1 - \)\(63\!\cdots\!28\)\( T + \)\(14\!\cdots\!02\)\( T^{2} - \)\(28\!\cdots\!64\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(20\!\cdots\!52\)\( T^{5} + \)\(57\!\cdots\!92\)\( T^{6} - \)\(20\!\cdots\!52\)\( p^{43} T^{7} + \)\(11\!\cdots\!95\)\( p^{86} T^{8} - \)\(28\!\cdots\!64\)\( p^{129} T^{9} + \)\(14\!\cdots\!02\)\( p^{172} T^{10} - \)\(63\!\cdots\!28\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 73 | \( 1 + \)\(94\!\cdots\!24\)\( T + \)\(37\!\cdots\!46\)\( T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(87\!\cdots\!27\)\( T^{4} + \)\(96\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!56\)\( T^{6} + \)\(96\!\cdots\!60\)\( p^{43} T^{7} + \)\(87\!\cdots\!27\)\( p^{86} T^{8} + \)\(46\!\cdots\!40\)\( p^{129} T^{9} + \)\(37\!\cdots\!46\)\( p^{172} T^{10} + \)\(94\!\cdots\!24\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 79 | \( 1 - \)\(17\!\cdots\!16\)\( p T + \)\(98\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!96\)\( T^{3} + \)\(34\!\cdots\!87\)\( T^{4} + \)\(26\!\cdots\!72\)\( T^{5} + \)\(80\!\cdots\!24\)\( T^{6} + \)\(26\!\cdots\!72\)\( p^{43} T^{7} + \)\(34\!\cdots\!87\)\( p^{86} T^{8} + \)\(17\!\cdots\!96\)\( p^{129} T^{9} + \)\(98\!\cdots\!50\)\( p^{172} T^{10} - \)\(17\!\cdots\!16\)\( p^{216} T^{11} + p^{258} T^{12} \) | |
| 83 | \( 1 + \)\(50\!\cdots\!56\)\( T + \)\(25\!\cdots\!38\)\( T^{2} + \)\(80\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!47\)\( T^{4} + \)\(52\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} + \)\(52\!\cdots\!88\)\( p^{43} T^{7} + \)\(23\!\cdots\!47\)\( p^{86} T^{8} + \)\(80\!\cdots\!52\)\( p^{129} T^{9} + \)\(25\!\cdots\!38\)\( p^{172} T^{10} + \)\(50\!\cdots\!56\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 89 | \( 1 + \)\(85\!\cdots\!80\)\( T + \)\(27\!\cdots\!14\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!15\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(26\!\cdots\!80\)\( T^{6} + \)\(18\!\cdots\!00\)\( p^{43} T^{7} + \)\(33\!\cdots\!15\)\( p^{86} T^{8} + \)\(18\!\cdots\!00\)\( p^{129} T^{9} + \)\(27\!\cdots\!14\)\( p^{172} T^{10} + \)\(85\!\cdots\!80\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| 97 | \( 1 + \)\(21\!\cdots\!72\)\( T + \)\(10\!\cdots\!14\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!27\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!64\)\( T^{6} - \)\(19\!\cdots\!00\)\( p^{43} T^{7} + \)\(40\!\cdots\!27\)\( p^{86} T^{8} - \)\(23\!\cdots\!40\)\( p^{129} T^{9} + \)\(10\!\cdots\!14\)\( p^{172} T^{10} + \)\(21\!\cdots\!72\)\( p^{215} T^{11} + p^{258} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.66295971078637636312157466079, −4.77108568931663740401236759701, −4.48505997558367458929751178496, −4.08062247379992267029222191952, −3.97478506624596209783766212675, −3.87884067814848453592917695315, −3.86420449825924237512449089609, −3.12616234156629445995470037707, −2.94439445683128934698686576459, −2.92676754272271215265814000162, −2.88754896170029396806164853335, −2.81575458833857968633934582485, −2.63295322045674631639294614853, −1.99765969348392813648340468283, −1.93808950544337647457518230627, −1.92828783352303092371465194972, −1.55540465416900536560211907597, −1.53388693768990336573572175316, −1.23276145491799561175459829825, −1.02681355441409533244726668250, −0.67817876087817917893289549864, −0.61158599481884053384967345062, −0.53393815651952823889618033711, −0.44019237250485438074065282989, −0.14013708875378170334066339085, 0.14013708875378170334066339085, 0.44019237250485438074065282989, 0.53393815651952823889618033711, 0.61158599481884053384967345062, 0.67817876087817917893289549864, 1.02681355441409533244726668250, 1.23276145491799561175459829825, 1.53388693768990336573572175316, 1.55540465416900536560211907597, 1.92828783352303092371465194972, 1.93808950544337647457518230627, 1.99765969348392813648340468283, 2.63295322045674631639294614853, 2.81575458833857968633934582485, 2.88754896170029396806164853335, 2.92676754272271215265814000162, 2.94439445683128934698686576459, 3.12616234156629445995470037707, 3.86420449825924237512449089609, 3.87884067814848453592917695315, 3.97478506624596209783766212675, 4.08062247379992267029222191952, 4.48505997558367458929751178496, 4.77108568931663740401236759701, 5.66295971078637636312157466079
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.