Dirichlet series
| L(s) = 1 | − 6·2-s + 5·4-s − 43·5-s + 8·7-s + 40·8-s + 258·10-s + 28·11-s + 56·13-s − 48·14-s − 141·16-s − 19·17-s − 75·19-s − 215·20-s − 168·22-s − 131·23-s + 602·25-s − 336·26-s + 40·28-s − 515·29-s + 237·31-s + 206·32-s + 114·34-s − 344·35-s + 269·37-s + 450·38-s − 1.72e3·40-s − 471·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 5/8·4-s − 3.84·5-s + 0.431·7-s + 1.76·8-s + 8.15·10-s + 0.767·11-s + 1.19·13-s − 0.916·14-s − 2.20·16-s − 0.271·17-s − 0.905·19-s − 2.40·20-s − 1.62·22-s − 1.18·23-s + 4.81·25-s − 2.53·26-s + 0.269·28-s − 3.29·29-s + 1.37·31-s + 1.13·32-s + 0.575·34-s − 1.66·35-s + 1.19·37-s + 1.92·38-s − 6.79·40-s − 1.79·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 43^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.41730\times 10^{8}\) |
| Root analytic conductor: | \(4.77846\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 3^{12} \cdot 43^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | 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 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + 3 p T + 31 T^{2} + 29 p^{2} T^{3} + 221 p T^{4} + 23 p^{6} T^{5} + 1167 p^{2} T^{6} + 23 p^{9} T^{7} + 221 p^{7} T^{8} + 29 p^{11} T^{9} + 31 p^{12} T^{10} + 3 p^{16} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 + 43 T + 1247 T^{2} + 26367 T^{3} + 452519 T^{4} + 6421366 T^{5} + 77975134 T^{6} + 6421366 p^{3} T^{7} + 452519 p^{6} T^{8} + 26367 p^{9} T^{9} + 1247 p^{12} T^{10} + 43 p^{15} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 - 8 T + 886 T^{2} - 4080 T^{3} + 63165 p T^{4} - 3221432 T^{5} + 186242508 T^{6} - 3221432 p^{3} T^{7} + 63165 p^{7} T^{8} - 4080 p^{9} T^{9} + 886 p^{12} T^{10} - 8 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 28 T + 3144 T^{2} - 41080 T^{3} + 3977536 T^{4} - 285452 p T^{5} + 4111332998 T^{6} - 285452 p^{4} T^{7} + 3977536 p^{6} T^{8} - 41080 p^{9} T^{9} + 3144 p^{12} T^{10} - 28 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 2150765192 p^{3} T^{7} + 37473880 p^{6} T^{8} - 530884 p^{9} T^{9} + 8776 p^{12} T^{10} - 56 p^{15} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 19 T + 23143 T^{2} + 543393 T^{3} + 241535186 T^{4} + 5650313095 T^{5} + 1493450611759 T^{6} + 5650313095 p^{3} T^{7} + 241535186 p^{6} T^{8} + 543393 p^{9} T^{9} + 23143 p^{12} T^{10} + 19 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 + 75 T + 38259 T^{2} + 2365781 T^{3} + 629552155 T^{4} + 31151517862 T^{5} + 5681041321490 T^{6} + 31151517862 p^{3} T^{7} + 629552155 p^{6} T^{8} + 2365781 p^{9} T^{9} + 38259 p^{12} T^{10} + 75 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 + 131 T + 52195 T^{2} + 3677795 T^{3} + 974730114 T^{4} + 33210519163 T^{5} + 11858751245947 T^{6} + 33210519163 p^{3} T^{7} + 974730114 p^{6} T^{8} + 3677795 p^{9} T^{9} + 52195 p^{12} T^{10} + 131 p^{15} T^{11} + p^{18} T^{12} \) | |
| 29 | \( 1 + 515 T + 204583 T^{2} + 58068807 T^{3} + 13900558631 T^{4} + 2733986934494 T^{5} + 464005745217070 T^{6} + 2733986934494 p^{3} T^{7} + 13900558631 p^{6} T^{8} + 58068807 p^{9} T^{9} + 204583 p^{12} T^{10} + 515 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 237 T + 125373 T^{2} - 21016589 T^{3} + 7321362670 T^{4} - 1031063540341 T^{5} + 275109610824401 T^{6} - 1031063540341 p^{3} T^{7} + 7321362670 p^{6} T^{8} - 21016589 p^{9} T^{9} + 125373 p^{12} T^{10} - 237 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 1302357216602 p^{3} T^{7} + 8177560635 p^{6} T^{8} - 30748693 p^{9} T^{9} + 126311 p^{12} T^{10} - 269 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 471 T + 349763 T^{2} + 112600045 T^{3} + 50459129866 T^{4} + 12922113800443 T^{5} + 4372223136871043 T^{6} + 12922113800443 p^{3} T^{7} + 50459129866 p^{6} T^{8} + 112600045 p^{9} T^{9} + 349763 p^{12} T^{10} + 471 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 + 415 T + 421631 T^{2} + 116866317 T^{3} + 77411983523 T^{4} + 16052264514750 T^{5} + 9154052369892234 T^{6} + 16052264514750 p^{3} T^{7} + 77411983523 p^{6} T^{8} + 116866317 p^{9} T^{9} + 421631 p^{12} T^{10} + 415 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 + 450 T + 321704 T^{2} + 149982378 T^{3} + 77929548632 T^{4} + 28654270442506 T^{5} + 12746558079363422 T^{6} + 28654270442506 p^{3} T^{7} + 77929548632 p^{6} T^{8} + 149982378 p^{9} T^{9} + 321704 p^{12} T^{10} + 450 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 + 356 T + 1153950 T^{2} + 347607228 T^{3} + 570632518215 T^{4} + 139273869185096 T^{5} + 154351054402988548 T^{6} + 139273869185096 p^{3} T^{7} + 570632518215 p^{6} T^{8} + 347607228 p^{9} T^{9} + 1153950 p^{12} T^{10} + 356 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} + 649067768079368 p^{3} T^{7} + 1136828745699 p^{6} T^{8} + 1454429624 p^{9} T^{9} + 1795994 p^{12} T^{10} + 1328 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 + 632 T + 927628 T^{2} + 253029932 T^{3} + 179674044568 T^{4} - 62778009874096 T^{5} - 5212390617577006 T^{6} - 62778009874096 p^{3} T^{7} + 179674044568 p^{6} T^{8} + 253029932 p^{9} T^{9} + 927628 p^{12} T^{10} + 632 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 144 T + 863230 T^{2} - 72744496 T^{3} + 475313592223 T^{4} - 62333254020128 T^{5} + 209643801201434276 T^{6} - 62333254020128 p^{3} T^{7} + 475313592223 p^{6} T^{8} - 72744496 p^{9} T^{9} + 863230 p^{12} T^{10} - 144 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 209583356925592 p^{3} T^{7} + 458263245867 p^{6} T^{8} - 439706488 p^{9} T^{9} + 1080658 p^{12} T^{10} - 864 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 + 1613 T + 2731081 T^{2} + 3130876751 T^{3} + 3268965374139 T^{4} + 2799662350208018 T^{5} + 2111366737833161318 T^{6} + 2799662350208018 p^{3} T^{7} + 3268965374139 p^{6} T^{8} + 3130876751 p^{9} T^{9} + 2731081 p^{12} T^{10} + 1613 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 682 T + 1120004 T^{2} - 1783287782 T^{3} + 1291656570448 T^{4} - 1149121488958882 T^{5} + 1227368803599345682 T^{6} - 1149121488958882 p^{3} T^{7} + 1291656570448 p^{6} T^{8} - 1783287782 p^{9} T^{9} + 1120004 p^{12} T^{10} - 682 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 3378 T + 7851850 T^{2} + 13055260850 T^{3} + 17370332054203 T^{4} + 19017874978895668 T^{5} + 17369747195795800052 T^{6} + 19017874978895668 p^{3} T^{7} + 17370332054203 p^{6} T^{8} + 13055260850 p^{9} T^{9} + 7851850 p^{12} T^{10} + 3378 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} - 1706175158728421 p^{3} T^{7} + 3317883854770 p^{6} T^{8} - 940317011 p^{9} T^{9} + 2496871 p^{12} T^{10} + 55 p^{15} T^{11} + p^{18} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.33457329404535877078162844225, −6.28317489401454061446700187726, −5.85250564690975039374830970678, −5.81871872892548333193166914547, −5.42900463802638485777985861034, −5.24830676940554805169476690130, −5.07113089963086051500087371127, −4.68542259197288803088603090618, −4.56998125627425664982277679250, −4.55596103853211547216056477973, −4.22070305699975927189122420075, −4.10802888478673101839331560823, −4.04045745152049818970268929744, −3.76366635061339841868335282523, −3.51256224523793588938939565594, −3.49631772126000274260379887345, −3.39783154111978565925044862582, −3.03664285376398048912581329320, −2.63430226185282897703441718412, −2.22891666055076259276258524745, −2.15364535380014020143929520484, −1.52957358217114384868025240807, −1.36771069855019565016563942882, −1.35879314000065357666312578326, −1.12044794649591594134142839369, 0, 0, 0, 0, 0, 0, 1.12044794649591594134142839369, 1.35879314000065357666312578326, 1.36771069855019565016563942882, 1.52957358217114384868025240807, 2.15364535380014020143929520484, 2.22891666055076259276258524745, 2.63430226185282897703441718412, 3.03664285376398048912581329320, 3.39783154111978565925044862582, 3.49631772126000274260379887345, 3.51256224523793588938939565594, 3.76366635061339841868335282523, 4.04045745152049818970268929744, 4.10802888478673101839331560823, 4.22070305699975927189122420075, 4.55596103853211547216056477973, 4.56998125627425664982277679250, 4.68542259197288803088603090618, 5.07113089963086051500087371127, 5.24830676940554805169476690130, 5.42900463802638485777985861034, 5.81871872892548333193166914547, 5.85250564690975039374830970678, 6.28317489401454061446700187726, 6.33457329404535877078162844225
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.