Dirichlet series
| L(s) = 1 | + 3·4-s − 12·9-s + 6·11-s − 12·13-s + 3·16-s + 12·17-s + 36·19-s + 6·23-s − 6·25-s − 4·27-s + 6·29-s − 24·31-s − 36·36-s + 18·41-s + 18·44-s + 12·47-s + 21·49-s − 36·52-s + 18·53-s + 18·59-s + 18·61-s − 2·64-s + 12·67-s + 36·68-s + 18·71-s + 108·76-s + 75·81-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 4·9-s + 1.80·11-s − 3.32·13-s + 3/4·16-s + 2.91·17-s + 8.25·19-s + 1.25·23-s − 6/5·25-s − 0.769·27-s + 1.11·29-s − 4.31·31-s − 6·36-s + 2.81·41-s + 2.71·44-s + 1.75·47-s + 3·49-s − 4.99·52-s + 2.47·53-s + 2.34·59-s + 2.30·61-s − 1/4·64-s + 1.46·67-s + 4.36·68-s + 2.13·71-s + 12.3·76-s + 25/3·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 5^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.56547\) |
| Root analytic conductor: | \(1.01884\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 5^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.991826066\) |
| \(L(\frac12)\) | \(\approx\) | \(1.991826066\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 5 | \( 1 + 6 T^{2} + 8 T^{3} - 9 T^{4} + 24 T^{5} - 124 T^{6} + 24 p T^{7} - 9 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \) | |
| 13 | \( 1 + 12 T + 69 T^{2} + 202 T^{3} - 24 T^{4} - 3576 T^{5} - 18615 T^{6} - 3576 p T^{7} - 24 p^{2} T^{8} + 202 p^{3} T^{9} + 69 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 + 4 p T^{2} + 4 T^{3} + 23 p T^{4} + 14 p T^{5} + 260 T^{6} + 64 p T^{7} + 230 p T^{8} + 602 T^{9} + 134 p^{2} T^{10} + 202 p^{2} T^{11} + 2065 T^{12} + 202 p^{3} T^{13} + 134 p^{4} T^{14} + 602 p^{3} T^{15} + 230 p^{5} T^{16} + 64 p^{6} T^{17} + 260 p^{6} T^{18} + 14 p^{8} T^{19} + 23 p^{9} T^{20} + 4 p^{9} T^{21} + 4 p^{11} T^{22} + p^{12} T^{24} \) |
| 7 | \( 1 - 3 p T^{2} - 64 T^{3} + 267 T^{4} + 1296 T^{5} - 68 p T^{6} - 16272 T^{7} - 20835 T^{8} + 15632 p T^{9} + 52839 p T^{10} - 344688 T^{11} - 3170306 T^{12} - 344688 p T^{13} + 52839 p^{3} T^{14} + 15632 p^{4} T^{15} - 20835 p^{4} T^{16} - 16272 p^{5} T^{17} - 68 p^{7} T^{18} + 1296 p^{7} T^{19} + 267 p^{8} T^{20} - 64 p^{9} T^{21} - 3 p^{11} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 - 6 T + 9 T^{2} + 38 T^{3} - 201 T^{4} + 558 T^{5} - 1060 T^{6} + 1416 T^{7} - 333 T^{8} + 27290 T^{9} - 5019 p T^{10} - 903420 T^{11} + 5737174 T^{12} - 903420 p T^{13} - 5019 p^{3} T^{14} + 27290 p^{3} T^{15} - 333 p^{4} T^{16} + 1416 p^{5} T^{17} - 1060 p^{6} T^{18} + 558 p^{7} T^{19} - 201 p^{8} T^{20} + 38 p^{9} T^{21} + 9 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 12 T + 30 T^{2} + 236 T^{3} - 2061 T^{4} + 7470 T^{5} - 3742 T^{6} - 150156 T^{7} + 1005000 T^{8} - 3137842 T^{9} + 754044 T^{10} + 3374598 p T^{11} - 358460075 T^{12} + 3374598 p^{2} T^{13} + 754044 p^{2} T^{14} - 3137842 p^{3} T^{15} + 1005000 p^{4} T^{16} - 150156 p^{5} T^{17} - 3742 p^{6} T^{18} + 7470 p^{7} T^{19} - 2061 p^{8} T^{20} + 236 p^{9} T^{21} + 30 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 36 T + 33 p T^{2} - 6996 T^{3} + 56109 T^{4} - 346014 T^{5} + 1726182 T^{6} - 7242900 T^{7} + 25603683 T^{8} - 67514124 T^{9} + 46500387 T^{10} + 785555922 T^{11} - 5211604870 T^{12} + 785555922 p T^{13} + 46500387 p^{2} T^{14} - 67514124 p^{3} T^{15} + 25603683 p^{4} T^{16} - 7242900 p^{5} T^{17} + 1726182 p^{6} T^{18} - 346014 p^{7} T^{19} + 56109 p^{8} T^{20} - 6996 p^{9} T^{21} + 33 p^{11} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 6 T + 66 T^{2} - 488 T^{3} + 3021 T^{4} - 20760 T^{5} + 114806 T^{6} - 707166 T^{7} + 3760734 T^{8} - 910870 p T^{9} + 110742342 T^{10} - 542176020 T^{11} + 2811377041 T^{12} - 542176020 p T^{13} + 110742342 p^{2} T^{14} - 910870 p^{4} T^{15} + 3760734 p^{4} T^{16} - 707166 p^{5} T^{17} + 114806 p^{6} T^{18} - 20760 p^{7} T^{19} + 3021 p^{8} T^{20} - 488 p^{9} T^{21} + 66 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T + 132 T^{2} - 720 T^{3} + 8553 T^{4} - 49548 T^{5} + 15228 p T^{6} - 2605038 T^{7} + 19782054 T^{8} - 106656714 T^{9} + 734913156 T^{10} - 3588829356 T^{11} + 23029455989 T^{12} - 3588829356 p T^{13} + 734913156 p^{2} T^{14} - 106656714 p^{3} T^{15} + 19782054 p^{4} T^{16} - 2605038 p^{5} T^{17} + 15228 p^{7} T^{18} - 49548 p^{7} T^{19} + 8553 p^{8} T^{20} - 720 p^{9} T^{21} + 132 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 24 T + 288 T^{2} + 2288 T^{3} + 13110 T^{4} + 60420 T^{5} + 291872 T^{6} + 1952988 T^{7} + 15720195 T^{8} + 114579516 T^{9} + 717768 p^{2} T^{10} + 116493996 p T^{11} + 19046422352 T^{12} + 116493996 p^{2} T^{13} + 717768 p^{4} T^{14} + 114579516 p^{3} T^{15} + 15720195 p^{4} T^{16} + 1952988 p^{5} T^{17} + 291872 p^{6} T^{18} + 60420 p^{7} T^{19} + 13110 p^{8} T^{20} + 2288 p^{9} T^{21} + 288 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 129 T^{2} - 44 T^{3} + 8709 T^{4} + 7206 T^{5} - 350846 T^{6} - 790824 T^{7} + 8624199 T^{8} + 38484952 T^{9} - 89392101 T^{10} - 713801802 T^{11} + 15505378 T^{12} - 713801802 p T^{13} - 89392101 p^{2} T^{14} + 38484952 p^{3} T^{15} + 8624199 p^{4} T^{16} - 790824 p^{5} T^{17} - 350846 p^{6} T^{18} + 7206 p^{7} T^{19} + 8709 p^{8} T^{20} - 44 p^{9} T^{21} - 129 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 18 T + 126 T^{2} - 520 T^{3} + 1377 T^{4} - 576 T^{5} + 37514 T^{6} - 458478 T^{7} + 1963602 T^{8} + 195578 T^{9} - 159010974 T^{10} + 1237236084 T^{11} - 5256961103 T^{12} + 1237236084 p T^{13} - 159010974 p^{2} T^{14} + 195578 p^{3} T^{15} + 1963602 p^{4} T^{16} - 458478 p^{5} T^{17} + 37514 p^{6} T^{18} - 576 p^{7} T^{19} + 1377 p^{8} T^{20} - 520 p^{9} T^{21} + 126 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 36 T^{2} + 200 T^{3} + 453 T^{4} - 48 T^{5} + 19244 T^{6} + 292536 T^{7} - 114714 T^{8} + 5310336 T^{9} - 1895028 T^{10} - 214231368 T^{11} + 8811791633 T^{12} - 214231368 p T^{13} - 1895028 p^{2} T^{14} + 5310336 p^{3} T^{15} - 114714 p^{4} T^{16} + 292536 p^{5} T^{17} + 19244 p^{6} T^{18} - 48 p^{7} T^{19} + 453 p^{8} T^{20} + 200 p^{9} T^{21} - 36 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 6 T + 177 T^{2} - 1214 T^{3} + 15390 T^{4} - 104838 T^{5} + 866273 T^{6} - 104838 p T^{7} + 15390 p^{2} T^{8} - 1214 p^{3} T^{9} + 177 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 18 T + 162 T^{2} - 752 T^{3} + 1449 T^{4} - 10704 T^{5} + 240686 T^{6} - 2011902 T^{7} + 13225086 T^{8} - 110699342 T^{9} + 1071608322 T^{10} - 8239654956 T^{11} + 58763275657 T^{12} - 8239654956 p T^{13} + 1071608322 p^{2} T^{14} - 110699342 p^{3} T^{15} + 13225086 p^{4} T^{16} - 2011902 p^{5} T^{17} + 240686 p^{6} T^{18} - 10704 p^{7} T^{19} + 1449 p^{8} T^{20} - 752 p^{9} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 18 T - 6 T^{2} + 1560 T^{3} - 7143 T^{4} + 34836 T^{5} - 171810 T^{6} - 6258942 T^{7} + 55320090 T^{8} - 174576126 T^{9} + 1622940978 T^{10} + 16173286548 T^{11} - 366681954847 T^{12} + 16173286548 p T^{13} + 1622940978 p^{2} T^{14} - 174576126 p^{3} T^{15} + 55320090 p^{4} T^{16} - 6258942 p^{5} T^{17} - 171810 p^{6} T^{18} + 34836 p^{7} T^{19} - 7143 p^{8} T^{20} + 1560 p^{9} T^{21} - 6 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 18 T - 72 T^{2} + 1984 T^{3} + 15813 T^{4} - 182406 T^{5} - 2198060 T^{6} + 13223814 T^{7} + 206194278 T^{8} - 603469244 T^{9} - 16276832034 T^{10} + 15373816002 T^{11} + 1051161809293 T^{12} + 15373816002 p T^{13} - 16276832034 p^{2} T^{14} - 603469244 p^{3} T^{15} + 206194278 p^{4} T^{16} + 13223814 p^{5} T^{17} - 2198060 p^{6} T^{18} - 182406 p^{7} T^{19} + 15813 p^{8} T^{20} + 1984 p^{9} T^{21} - 72 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 12 T + 390 T^{2} - 4104 T^{3} + 79149 T^{4} - 717432 T^{5} + 10567682 T^{6} - 84269268 T^{7} + 1049328186 T^{8} - 7501086108 T^{9} + 84719286510 T^{10} - 560297874168 T^{11} + 5970730533525 T^{12} - 560297874168 p T^{13} + 84719286510 p^{2} T^{14} - 7501086108 p^{3} T^{15} + 1049328186 p^{4} T^{16} - 84269268 p^{5} T^{17} + 10567682 p^{6} T^{18} - 717432 p^{7} T^{19} + 79149 p^{8} T^{20} - 4104 p^{9} T^{21} + 390 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 18 T + 198 T^{2} - 784 T^{3} - 4815 T^{4} + 165420 T^{5} - 1730470 T^{6} + 15633210 T^{7} - 88800294 T^{8} + 322869362 T^{9} + 5177337294 T^{10} - 94262889492 T^{11} + 1094451263737 T^{12} - 94262889492 p T^{13} + 5177337294 p^{2} T^{14} + 322869362 p^{3} T^{15} - 88800294 p^{4} T^{16} + 15633210 p^{5} T^{17} - 1730470 p^{6} T^{18} + 165420 p^{7} T^{19} - 4815 p^{8} T^{20} - 784 p^{9} T^{21} + 198 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 576 T^{2} + 165150 T^{4} - 30997624 T^{6} + 4232770083 T^{8} - 442599153120 T^{10} + 36330094124880 T^{12} - 442599153120 p^{2} T^{14} + 4232770083 p^{4} T^{16} - 30997624 p^{6} T^{18} + 165150 p^{8} T^{20} - 576 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( 1 - 504 T^{2} + 130482 T^{4} - 22867736 T^{6} + 38366469 p T^{8} - 321022992024 T^{10} + 27891029278968 T^{12} - 321022992024 p^{2} T^{14} + 38366469 p^{5} T^{16} - 22867736 p^{6} T^{18} + 130482 p^{8} T^{20} - 504 p^{10} T^{22} + p^{12} T^{24} \) | |
| 83 | \( ( 1 + 24 T + 534 T^{2} + 7494 T^{3} + 1173 p T^{4} + 997746 T^{5} + 9916666 T^{6} + 997746 p T^{7} + 1173 p^{3} T^{8} + 7494 p^{3} T^{9} + 534 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 - 6 T + 159 T^{2} - 2026 T^{3} + 12543 T^{4} - 185880 T^{5} + 1109408 T^{6} - 4798980 T^{7} + 53180853 T^{8} - 218453974 T^{9} + 979750449 T^{10} - 21272322174 T^{11} + 157717614022 T^{12} - 21272322174 p T^{13} + 979750449 p^{2} T^{14} - 218453974 p^{3} T^{15} + 53180853 p^{4} T^{16} - 4798980 p^{5} T^{17} + 1109408 p^{6} T^{18} - 185880 p^{7} T^{19} + 12543 p^{8} T^{20} - 2026 p^{9} T^{21} + 159 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 102 T + 5478 T^{2} + 205020 T^{3} + 5971839 T^{4} + 143322978 T^{5} + 2931598594 T^{6} + 52233202278 T^{7} + 822698092296 T^{8} + 11571385503276 T^{9} + 146348424797580 T^{10} + 1671892202819538 T^{11} + 17296294921153857 T^{12} + 1671892202819538 p T^{13} + 146348424797580 p^{2} T^{14} + 11571385503276 p^{3} T^{15} + 822698092296 p^{4} T^{16} + 52233202278 p^{5} T^{17} + 2931598594 p^{6} T^{18} + 143322978 p^{7} T^{19} + 5971839 p^{8} T^{20} + 205020 p^{9} T^{21} + 5478 p^{10} T^{22} + 102 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−4.99890818283639070088238985832, −4.82808486642148436201888897366, −4.71910771247798785302544792578, −4.30616271687869537022993357698, −4.19853590948820991476716379269, −4.06416941277506729618634660186, −3.99381184012509492380932240858, −3.82513555513238304471628745603, −3.72092262287735982649051841937, −3.64251914335828298215627131226, −3.64026304453291543733098431316, −3.16793388237779430655519233887, −3.05061419569195185343293834072, −3.04539779753628967669268379559, −2.97710013482101789024940129080, −2.83214390743676487075129984002, −2.56550957275573053569205643755, −2.53663956082126172919142304496, −2.39436040980688737574769181107, −2.37304748355670007401130697669, −2.09784444105126655131885217082, −1.41512676808580650492133957501, −1.20274576890857845419593956663, −1.10453740983235225640494206554, −1.05326644270254489554451018446, 1.05326644270254489554451018446, 1.10453740983235225640494206554, 1.20274576890857845419593956663, 1.41512676808580650492133957501, 2.09784444105126655131885217082, 2.37304748355670007401130697669, 2.39436040980688737574769181107, 2.53663956082126172919142304496, 2.56550957275573053569205643755, 2.83214390743676487075129984002, 2.97710013482101789024940129080, 3.04539779753628967669268379559, 3.05061419569195185343293834072, 3.16793388237779430655519233887, 3.64026304453291543733098431316, 3.64251914335828298215627131226, 3.72092262287735982649051841937, 3.82513555513238304471628745603, 3.99381184012509492380932240858, 4.06416941277506729618634660186, 4.19853590948820991476716379269, 4.30616271687869537022993357698, 4.71910771247798785302544792578, 4.82808486642148436201888897366, 4.99890818283639070088238985832
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.