Dirichlet series
| L(s) = 1 | + 9·3-s − 9·5-s − 13·7-s + 51·9-s + 18·11-s − 5·13-s − 81·15-s − 562·19-s − 117·21-s + 1.71e3·23-s − 1.03e3·25-s + 2.11e3·29-s − 187·31-s + 162·33-s + 117·35-s + 16·37-s − 45·39-s − 7.92e3·41-s + 68·43-s − 459·45-s − 1.36e4·47-s + 4.72e3·49-s − 162·55-s − 5.05e3·57-s + 2.00e4·59-s − 1.93e3·61-s − 663·63-s + ⋯ |
| L(s) = 1 | + 3-s − 0.359·5-s − 0.265·7-s + 0.629·9-s + 0.148·11-s − 0.0295·13-s − 0.359·15-s − 1.55·19-s − 0.265·21-s + 3.24·23-s − 1.65·25-s + 2.51·29-s − 0.194·31-s + 0.148·33-s + 0.0955·35-s + 0.0116·37-s − 0.0295·39-s − 4.71·41-s + 0.0367·43-s − 0.226·45-s − 6.19·47-s + 1.96·49-s − 0.0535·55-s − 1.55·57-s + 5.76·59-s − 0.520·61-s − 0.167·63-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.41020\times 10^{9}\) |
| Root analytic conductor: | \(3.85814\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{16} ,\ ( \ : [2]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.3734065759\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3734065759\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T + 10 p T^{2} + 7 p^{3} T^{3} - 86 p^{4} T^{4} + 7 p^{7} T^{5} + 10 p^{9} T^{6} - p^{14} T^{7} + p^{16} T^{8} \) | |
| good | 5 | \( 1 + 9 T + 1114 T^{2} + 9783 T^{3} + 533599 T^{4} + 10528056 T^{5} + 59806456 T^{6} + 10305069192 T^{7} - 6001445444 T^{8} + 10305069192 p^{4} T^{9} + 59806456 p^{8} T^{10} + 10528056 p^{12} T^{11} + 533599 p^{16} T^{12} + 9783 p^{20} T^{13} + 1114 p^{24} T^{14} + 9 p^{28} T^{15} + p^{32} T^{16} \) |
| 7 | \( 1 + 13 T - 4554 T^{2} - 124753 T^{3} + 8962391 T^{4} + 7093476 p^{2} T^{5} + 2901735784 T^{6} - 447642835484 T^{7} - 30706182623268 T^{8} - 447642835484 p^{4} T^{9} + 2901735784 p^{8} T^{10} + 7093476 p^{14} T^{11} + 8962391 p^{16} T^{12} - 124753 p^{20} T^{13} - 4554 p^{24} T^{14} + 13 p^{28} T^{15} + p^{32} T^{16} \) | |
| 11 | \( 1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 445095012639474 p^{4} T^{9} + 9074477968558 p^{8} T^{10} - 17443950048 p^{12} T^{11} + 452208721 p^{16} T^{12} - 607356 p^{20} T^{13} + 33850 p^{24} T^{14} - 18 p^{28} T^{15} + p^{32} T^{16} \) | |
| 13 | \( 1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 3464858176098460 p^{4} T^{9} + 23191943061904 p^{8} T^{10} + 250092029040 p^{12} T^{11} + 352176575 p^{16} T^{12} - 7266665 p^{20} T^{13} - 42054 p^{24} T^{14} + 5 p^{28} T^{15} + p^{32} T^{16} \) | |
| 17 | \( 1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} + \)\(63\!\cdots\!86\)\( T^{8} - 6759733382202755 p^{8} T^{10} + 52320681154 p^{16} T^{12} - 288125 p^{24} T^{14} + p^{32} T^{16} \) | |
| 19 | \( ( 1 + 281 T + 110170 T^{2} - 68843041 T^{3} - 21846246566 T^{4} - 68843041 p^{4} T^{5} + 110170 p^{8} T^{6} + 281 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 23 | \( 1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} + \)\(10\!\cdots\!68\)\( T^{7} - \)\(65\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!68\)\( p^{4} T^{9} - 118516645434237428 p^{8} T^{10} + 16333135609500 p^{12} T^{11} + 342642958747 p^{16} T^{12} - 935945649 p^{20} T^{13} + 1529458 p^{24} T^{14} - 1719 p^{28} T^{15} + p^{32} T^{16} \) | |
| 29 | \( 1 - 2115 T + 4091014 T^{2} - 5498870985 T^{3} + 7048695081595 T^{4} - 8024737206821040 T^{5} + 8297140249856169556 T^{6} - \)\(79\!\cdots\!80\)\( T^{7} + \)\(68\!\cdots\!24\)\( T^{8} - \)\(79\!\cdots\!80\)\( p^{4} T^{9} + 8297140249856169556 p^{8} T^{10} - 8024737206821040 p^{12} T^{11} + 7048695081595 p^{16} T^{12} - 5498870985 p^{20} T^{13} + 4091014 p^{24} T^{14} - 2115 p^{28} T^{15} + p^{32} T^{16} \) | |
| 31 | \( 1 + 187 T - 2516004 T^{2} - 186847537 T^{3} + 3362431719041 T^{4} + 9550301616 p T^{5} - 3460282108678916846 T^{6} + 13487262715981377034 T^{7} + \)\(31\!\cdots\!92\)\( T^{8} + 13487262715981377034 p^{4} T^{9} - 3460282108678916846 p^{8} T^{10} + 9550301616 p^{13} T^{11} + 3362431719041 p^{16} T^{12} - 186847537 p^{20} T^{13} - 2516004 p^{24} T^{14} + 187 p^{28} T^{15} + p^{32} T^{16} \) | |
| 37 | \( ( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 1256575624 p^{4} T^{5} + 3611368 p^{8} T^{6} - 8 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 41 | \( 1 + 7920 T + 37687894 T^{2} + 132890424480 T^{3} + 385083705354505 T^{4} + 963185727644706960 T^{5} + \)\(21\!\cdots\!66\)\( T^{6} + \)\(41\!\cdots\!20\)\( T^{7} + \)\(74\!\cdots\!64\)\( T^{8} + \)\(41\!\cdots\!20\)\( p^{4} T^{9} + \)\(21\!\cdots\!66\)\( p^{8} T^{10} + 963185727644706960 p^{12} T^{11} + 385083705354505 p^{16} T^{12} + 132890424480 p^{20} T^{13} + 37687894 p^{24} T^{14} + 7920 p^{28} T^{15} + p^{32} T^{16} \) | |
| 43 | \( 1 - 68 T - 12950604 T^{2} + 209786648 T^{3} + 102574547445791 T^{4} - 59145034173804 T^{5} - \)\(54\!\cdots\!36\)\( T^{6} - \)\(27\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} - \)\(27\!\cdots\!16\)\( p^{4} T^{9} - \)\(54\!\cdots\!36\)\( p^{8} T^{10} - 59145034173804 p^{12} T^{11} + 102574547445791 p^{16} T^{12} + 209786648 p^{20} T^{13} - 12950604 p^{24} T^{14} - 68 p^{28} T^{15} + p^{32} T^{16} \) | |
| 47 | \( 1 + 13689 T + 103685338 T^{2} + 12006252297 p T^{3} + 2435028217967227 T^{4} + 185888262507277500 p T^{5} + \)\(26\!\cdots\!72\)\( T^{6} + \)\(15\!\cdots\!56\)\( p T^{7} + \)\(76\!\cdots\!80\)\( p^{2} T^{8} + \)\(15\!\cdots\!56\)\( p^{5} T^{9} + \)\(26\!\cdots\!72\)\( p^{8} T^{10} + 185888262507277500 p^{13} T^{11} + 2435028217967227 p^{16} T^{12} + 12006252297 p^{21} T^{13} + 103685338 p^{24} T^{14} + 13689 p^{28} T^{15} + p^{32} T^{16} \) | |
| 53 | \( 1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} + \)\(67\!\cdots\!26\)\( T^{8} - 84051566001475463360 p^{8} T^{10} + 115452291970684 p^{16} T^{12} - 5145920 p^{24} T^{14} + p^{32} T^{16} \) | |
| 59 | \( 1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!08\)\( T^{8} - \)\(84\!\cdots\!76\)\( p^{4} T^{9} + \)\(21\!\cdots\!48\)\( p^{8} T^{10} - 49579528565018409012 p^{12} T^{11} + 9931594296358591 p^{16} T^{12} - 1657982214864 p^{20} T^{13} + 216711700 p^{24} T^{14} - 20052 p^{28} T^{15} + p^{32} T^{16} \) | |
| 61 | \( 1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} + \)\(12\!\cdots\!24\)\( T^{6} + \)\(40\!\cdots\!44\)\( T^{7} - \)\(98\!\cdots\!68\)\( T^{8} + \)\(40\!\cdots\!44\)\( p^{4} T^{9} + \)\(12\!\cdots\!24\)\( p^{8} T^{10} - 1350680282380662864 p^{12} T^{11} + 416288373490931 p^{16} T^{12} + 149647181023 p^{20} T^{13} - 10529634 p^{24} T^{14} + 1937 p^{28} T^{15} + p^{32} T^{16} \) | |
| 67 | \( 1 + 154 T - 33835854 T^{2} + 25606229228 T^{3} + 539365905411977 T^{4} - 738160924156362336 T^{5} + \)\(70\!\cdots\!02\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} + \)\(11\!\cdots\!78\)\( p^{4} T^{9} + \)\(70\!\cdots\!02\)\( p^{8} T^{10} - 738160924156362336 p^{12} T^{11} + 539365905411977 p^{16} T^{12} + 25606229228 p^{20} T^{13} - 33835854 p^{24} T^{14} + 154 p^{28} T^{15} + p^{32} T^{16} \) | |
| 71 | \( 1 - 68871716 T^{2} + 3244147638477940 T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(11\!\cdots\!24\)\( p^{8} T^{10} + 3244147638477940 p^{16} T^{12} - 68871716 p^{24} T^{14} + p^{32} T^{16} \) | |
| 73 | \( ( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 292589317519 p^{4} T^{5} + 59309470 p^{8} T^{6} + 3901 p^{12} T^{7} + p^{16} T^{8} )^{2} \) | |
| 79 | \( 1 - 2195 T - 87724914 T^{2} + 187644610415 T^{3} + 3128319215246375 T^{4} - 3711455091635884260 T^{5} - \)\(16\!\cdots\!36\)\( T^{6} - \)\(76\!\cdots\!80\)\( T^{7} + \)\(88\!\cdots\!64\)\( T^{8} - \)\(76\!\cdots\!80\)\( p^{4} T^{9} - \)\(16\!\cdots\!36\)\( p^{8} T^{10} - 3711455091635884260 p^{12} T^{11} + 3128319215246375 p^{16} T^{12} + 187644610415 p^{20} T^{13} - 87724914 p^{24} T^{14} - 2195 p^{28} T^{15} + p^{32} T^{16} \) | |
| 83 | \( 1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} + \)\(87\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!28\)\( T^{6} + \)\(40\!\cdots\!76\)\( T^{7} + \)\(26\!\cdots\!48\)\( T^{8} + \)\(40\!\cdots\!76\)\( p^{4} T^{9} + \)\(62\!\cdots\!28\)\( p^{8} T^{10} + \)\(87\!\cdots\!72\)\( p^{12} T^{11} + 104510585134438411 p^{16} T^{12} + 9956481997959 p^{20} T^{13} + 725723290 p^{24} T^{14} + 37017 p^{28} T^{15} + p^{32} T^{16} \) | |
| 89 | \( 1 - 294759296 T^{2} + 46567064448316540 T^{4} - \)\(48\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!14\)\( T^{8} - \)\(48\!\cdots\!04\)\( p^{8} T^{10} + 46567064448316540 p^{16} T^{12} - 294759296 p^{24} T^{14} + p^{32} T^{16} \) | |
| 97 | \( 1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(33\!\cdots\!26\)\( T^{7} + \)\(75\!\cdots\!52\)\( T^{8} + \)\(33\!\cdots\!26\)\( p^{4} T^{9} - \)\(75\!\cdots\!16\)\( p^{8} T^{10} - 97460573991801682656 p^{12} T^{11} + 57034963146137471 p^{16} T^{12} + 993163976152 p^{20} T^{13} - 283226964 p^{24} T^{14} - 7282 p^{28} T^{15} + p^{32} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.13065006412048629610695609082, −4.97345562376240289964541913163, −4.92041927068425765822392666872, −4.79116129179788715863154299711, −4.68388081871570893709196194262, −4.42869228188299747935743847390, −3.99045743608459420725970651985, −3.92617678284388735586544918010, −3.85972421433056856371340405219, −3.62378944663441614405976398267, −3.56866573136195323142931825986, −3.19843659037482583267581172097, −3.07774632852087515197614801894, −2.81078195202837834712878464653, −2.69851067928946843269307108473, −2.61310598730563139152861042956, −2.31437406652824639884788343315, −1.86068707783967183375034801165, −1.71660982039995266101706480963, −1.62208784451203667254856550991, −1.34914749089781402497938657969, −1.03804645440759770204875783834, −0.804915982229737448236203514581, −0.29660334446422853351632516589, −0.07143012137178784279314411600, 0.07143012137178784279314411600, 0.29660334446422853351632516589, 0.804915982229737448236203514581, 1.03804645440759770204875783834, 1.34914749089781402497938657969, 1.62208784451203667254856550991, 1.71660982039995266101706480963, 1.86068707783967183375034801165, 2.31437406652824639884788343315, 2.61310598730563139152861042956, 2.69851067928946843269307108473, 2.81078195202837834712878464653, 3.07774632852087515197614801894, 3.19843659037482583267581172097, 3.56866573136195323142931825986, 3.62378944663441614405976398267, 3.85972421433056856371340405219, 3.92617678284388735586544918010, 3.99045743608459420725970651985, 4.42869228188299747935743847390, 4.68388081871570893709196194262, 4.79116129179788715863154299711, 4.92041927068425765822392666872, 4.97345562376240289964541913163, 5.13065006412048629610695609082
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.