Dirichlet series
| L(s) = 1 | + 2·2-s + 3-s + 4-s − 5-s + 2·6-s + 3·7-s + 3·9-s − 2·10-s + 12-s + 8·13-s + 6·14-s − 15-s + 2·16-s + 8·17-s + 6·18-s − 20-s + 3·21-s + 28·23-s + 7·25-s + 16·26-s + 8·27-s + 3·28-s − 2·30-s + 13·31-s − 14·32-s + 16·34-s − 3·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 1.13·7-s + 9-s − 0.632·10-s + 0.288·12-s + 2.21·13-s + 1.60·14-s − 0.258·15-s + 1/2·16-s + 1.94·17-s + 1.41·18-s − 0.223·20-s + 0.654·21-s + 5.83·23-s + 7/5·25-s + 3.13·26-s + 1.53·27-s + 0.566·28-s − 0.365·30-s + 2.33·31-s − 2.47·32-s + 2.74·34-s − 0.507·35-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{12} \cdot 11^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.16064\times 10^{9}\) |
| Root analytic conductor: | \(2.60064\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{12} \cdot 11^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(139.9239619\) |
| \(L(\frac12)\) | \(\approx\) | \(139.9239619\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + 3 T^{4} + p^{4} T^{5} - 33 T^{6} + 25 p T^{7} - 65 T^{8} + 7 p^{3} T^{9} + 59 p T^{10} - p^{8} T^{11} + 97 p^{2} T^{12} - p^{9} T^{13} + 59 p^{3} T^{14} + 7 p^{6} T^{15} - 65 p^{4} T^{16} + 25 p^{6} T^{17} - 33 p^{6} T^{18} + p^{11} T^{19} + 3 p^{8} T^{20} - p^{11} T^{21} + 3 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) |
| 3 | \( 1 - T - 2 T^{2} - p T^{3} + 8 T^{4} - 32 T^{5} + 32 T^{6} + 8 p^{2} T^{7} + 97 T^{8} - 272 T^{9} + 712 T^{10} - 248 p T^{11} - 1601 T^{12} - 248 p^{2} T^{13} + 712 p^{2} T^{14} - 272 p^{3} T^{15} + 97 p^{4} T^{16} + 8 p^{7} T^{17} + 32 p^{6} T^{18} - 32 p^{7} T^{19} + 8 p^{8} T^{20} - p^{10} T^{21} - 2 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( 1 + T - 6 T^{2} - 7 T^{3} + 6 T^{4} + 28 T^{5} + 54 T^{6} - 136 T^{7} + 19 T^{8} + 268 p T^{9} + 3418 T^{10} - 3238 T^{11} - 29939 T^{12} - 3238 p T^{13} + 3418 p^{2} T^{14} + 268 p^{4} T^{15} + 19 p^{4} T^{16} - 136 p^{5} T^{17} + 54 p^{6} T^{18} + 28 p^{7} T^{19} + 6 p^{8} T^{20} - 7 p^{9} T^{21} - 6 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 8 T + 23 T^{2} - 324 T^{4} - 760 T^{5} + 15724 T^{6} - 61496 T^{7} + 78885 T^{8} + 469128 T^{9} - 888268 T^{10} - 10299304 T^{11} + 57810247 T^{12} - 10299304 p T^{13} - 888268 p^{2} T^{14} + 469128 p^{3} T^{15} + 78885 p^{4} T^{16} - 61496 p^{5} T^{17} + 15724 p^{6} T^{18} - 760 p^{7} T^{19} - 324 p^{8} T^{20} + 23 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 11 T^{2} + 128 T^{3} - 844 T^{4} - 1016 T^{5} + 29908 T^{6} - 111320 T^{7} - 72771 T^{8} + 2499144 T^{9} - 6443380 T^{10} - 29627592 T^{11} + 234698431 T^{12} - 29627592 p T^{13} - 6443380 p^{2} T^{14} + 2499144 p^{3} T^{15} - 72771 p^{4} T^{16} - 111320 p^{5} T^{17} + 29908 p^{6} T^{18} - 1016 p^{7} T^{19} - 844 p^{8} T^{20} + 128 p^{9} T^{21} + 11 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 17 T^{2} - 64 T^{3} - 34 T^{4} - 3264 T^{5} + 174 p T^{6} + 78336 T^{7} + 280279 T^{8} - 50048 T^{9} + 3108178 T^{10} - 20001792 T^{11} - 162841925 T^{12} - 20001792 p T^{13} + 3108178 p^{2} T^{14} - 50048 p^{3} T^{15} + 280279 p^{4} T^{16} + 78336 p^{5} T^{17} + 174 p^{7} T^{18} - 3264 p^{7} T^{19} - 34 p^{8} T^{20} - 64 p^{9} T^{21} - 17 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 7 T + 77 T^{2} - 314 T^{3} + 77 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 29 | \( 1 - 47 T^{2} + 64 T^{3} + 846 T^{4} + 9024 T^{5} + 4006 T^{6} - 457216 T^{7} + 382439 T^{8} + 7884928 T^{9} + 61357278 T^{10} + 56983552 T^{11} - 2998123365 T^{12} + 56983552 p T^{13} + 61357278 p^{2} T^{14} + 7884928 p^{3} T^{15} + 382439 p^{4} T^{16} - 457216 p^{5} T^{17} + 4006 p^{6} T^{18} + 9024 p^{7} T^{19} + 846 p^{8} T^{20} + 64 p^{9} T^{21} - 47 p^{10} T^{22} + p^{12} T^{24} \) | |
| 31 | \( 1 - 13 T + 26 T^{2} + 657 T^{3} - 5460 T^{4} + 15964 T^{5} + 22960 T^{6} - 713908 T^{7} + 7426341 T^{8} - 36420348 T^{9} + 3945344 T^{10} + 1142793028 T^{11} - 8639431253 T^{12} + 1142793028 p T^{13} + 3945344 p^{2} T^{14} - 36420348 p^{3} T^{15} + 7426341 p^{4} T^{16} - 713908 p^{5} T^{17} + 22960 p^{6} T^{18} + 15964 p^{7} T^{19} - 5460 p^{8} T^{20} + 657 p^{9} T^{21} + 26 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 17 T + 106 T^{2} + 35 T^{3} - 138 p T^{4} + 24264 T^{5} + 104582 T^{6} - 1555984 T^{7} + 4789839 T^{8} + 22847880 T^{9} - 147468394 T^{10} - 903867842 T^{11} + 11727294013 T^{12} - 903867842 p T^{13} - 147468394 p^{2} T^{14} + 22847880 p^{3} T^{15} + 4789839 p^{4} T^{16} - 1555984 p^{5} T^{17} + 104582 p^{6} T^{18} + 24264 p^{7} T^{19} - 138 p^{9} T^{20} + 35 p^{9} T^{21} + 106 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 16 T + 67 T^{2} + 560 T^{3} - 7100 T^{4} + 28000 T^{5} - 43388 T^{6} + 246672 T^{7} + 616013 T^{8} - 60687280 T^{9} + 471922140 T^{10} + 387365904 T^{11} - 19332075665 T^{12} + 387365904 p T^{13} + 471922140 p^{2} T^{14} - 60687280 p^{3} T^{15} + 616013 p^{4} T^{16} + 246672 p^{5} T^{17} - 43388 p^{6} T^{18} + 28000 p^{7} T^{19} - 7100 p^{8} T^{20} + 560 p^{9} T^{21} + 67 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 4 T + 101 T^{2} + 312 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 47 | \( 1 + 4 T - 111 T^{2} - 568 T^{3} + 7392 T^{4} + 49396 T^{5} - 365232 T^{6} - 2901388 T^{7} + 16129633 T^{8} + 122188172 T^{9} - 498402320 T^{10} - 2354973940 T^{11} + 16802495119 T^{12} - 2354973940 p T^{13} - 498402320 p^{2} T^{14} + 122188172 p^{3} T^{15} + 16129633 p^{4} T^{16} - 2901388 p^{5} T^{17} - 365232 p^{6} T^{18} + 49396 p^{7} T^{19} + 7392 p^{8} T^{20} - 568 p^{9} T^{21} - 111 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 10 T - 59 T^{2} + 1184 T^{3} - 926 T^{4} - 69722 T^{5} + 294406 T^{6} + 2232546 T^{7} - 16364317 T^{8} - 47271754 T^{9} + 615669366 T^{10} + 18015990 p T^{11} - 28997636753 T^{12} + 18015990 p^{2} T^{13} + 615669366 p^{2} T^{14} - 47271754 p^{3} T^{15} - 16364317 p^{4} T^{16} + 2232546 p^{5} T^{17} + 294406 p^{6} T^{18} - 69722 p^{7} T^{19} - 926 p^{8} T^{20} + 1184 p^{9} T^{21} - 59 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + T - 170 T^{2} - 221 T^{3} + 320 p T^{4} + 328 p T^{5} - 1722416 T^{6} - 1333488 T^{7} + 139186697 T^{8} + 76390936 T^{9} - 9605185704 T^{10} - 2482296192 T^{11} + 594022989415 T^{12} - 2482296192 p T^{13} - 9605185704 p^{2} T^{14} + 76390936 p^{3} T^{15} + 139186697 p^{4} T^{16} - 1333488 p^{5} T^{17} - 1722416 p^{6} T^{18} + 328 p^{8} T^{19} + 320 p^{9} T^{20} - 221 p^{9} T^{21} - 170 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 16 T + 7 T^{2} - 1840 T^{3} - 13860 T^{4} + 4320 T^{5} + 400652 T^{6} + 1904048 T^{7} + 38865093 T^{8} + 488718000 T^{9} + 1383027860 T^{10} - 26636392784 T^{11} - 345384133145 T^{12} - 26636392784 p T^{13} + 1383027860 p^{2} T^{14} + 488718000 p^{3} T^{15} + 38865093 p^{4} T^{16} + 1904048 p^{5} T^{17} + 400652 p^{6} T^{18} + 4320 p^{7} T^{19} - 13860 p^{8} T^{20} - 1840 p^{9} T^{21} + 7 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 3 T + 113 T^{2} - 22 T^{3} + 113 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 71 | \( 1 + 5 T + 20 T^{2} - 695 T^{3} - 7780 T^{4} - 68170 T^{5} - 5786 T^{6} + 5263790 T^{7} + 56924985 T^{8} + 339614790 T^{9} - 118215790 T^{10} - 23737080060 T^{11} - 265061379329 T^{12} - 23737080060 p T^{13} - 118215790 p^{2} T^{14} + 339614790 p^{3} T^{15} + 56924985 p^{4} T^{16} + 5263790 p^{5} T^{17} - 5786 p^{6} T^{18} - 68170 p^{7} T^{19} - 7780 p^{8} T^{20} - 695 p^{9} T^{21} + 20 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 16 T - 29 T^{2} + 2608 T^{3} - 15612 T^{4} - 54432 T^{5} + 597956 T^{6} - 1898096 T^{7} + 105180429 T^{8} - 1137548208 T^{9} + 139658396 T^{10} + 77531528720 T^{11} - 791045249105 T^{12} + 77531528720 p T^{13} + 139658396 p^{2} T^{14} - 1137548208 p^{3} T^{15} + 105180429 p^{4} T^{16} - 1898096 p^{5} T^{17} + 597956 p^{6} T^{18} - 54432 p^{7} T^{19} - 15612 p^{8} T^{20} + 2608 p^{9} T^{21} - 29 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 28 T + 319 T^{2} - 848 T^{3} - 23118 T^{4} + 368100 T^{5} - 2689762 T^{6} + 7469044 T^{7} + 64811091 T^{8} - 1075382940 T^{9} + 8371511678 T^{10} - 40089262324 T^{11} + 177958244563 T^{12} - 40089262324 p T^{13} + 8371511678 p^{2} T^{14} - 1075382940 p^{3} T^{15} + 64811091 p^{4} T^{16} + 7469044 p^{5} T^{17} - 2689762 p^{6} T^{18} + 368100 p^{7} T^{19} - 23118 p^{8} T^{20} - 848 p^{9} T^{21} + 319 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 8 T - 129 T^{2} + 1184 T^{3} + 10542 T^{4} - 294632 T^{5} + 609882 T^{6} + 30027704 T^{7} - 108165497 T^{8} - 2317255144 T^{9} + 27636921250 T^{10} + 30542384840 T^{11} - 2623552045061 T^{12} + 30542384840 p T^{13} + 27636921250 p^{2} T^{14} - 2317255144 p^{3} T^{15} - 108165497 p^{4} T^{16} + 30027704 p^{5} T^{17} + 609882 p^{6} T^{18} - 294632 p^{7} T^{19} + 10542 p^{8} T^{20} + 1184 p^{9} T^{21} - 129 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 21 T + 371 T^{2} + 3838 T^{3} + 371 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{4} \) | |
| 97 | \( 1 - 11 T - 138 T^{2} + 2685 T^{3} - 414 T^{4} - 183880 T^{5} + 1306526 T^{6} - 9183508 T^{7} - 69630505 T^{8} + 3158826216 T^{9} - 21575809142 T^{10} - 200368131722 T^{11} + 3986418389017 T^{12} - 200368131722 p T^{13} - 21575809142 p^{2} T^{14} + 3158826216 p^{3} T^{15} - 69630505 p^{4} T^{16} - 9183508 p^{5} T^{17} + 1306526 p^{6} T^{18} - 183880 p^{7} T^{19} - 414 p^{8} T^{20} + 2685 p^{9} T^{21} - 138 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.33447439434881564170680588041, −3.00095527361905711777154652304, −2.95609300377199411925271132969, −2.93768474949354172817029935327, −2.93682120733680784153308237835, −2.93306152307893602355891220264, −2.88864972284101754209685018802, −2.82664784027621376876732786567, −2.75176517843696163225953596956, −2.33123621712513767621339228602, −2.32472013418674254502329423353, −2.27980395868283393910415239732, −1.89209099181309468516671159340, −1.83525396397204908301591983170, −1.77570966479484114016352137639, −1.63035730219212311065457509288, −1.54165459308531058857531502027, −1.33090415261922327390826413942, −1.32562194773685813556297647417, −1.03685524306097078000162772402, −1.02051229791062010534719239452, −0.944765638933769613081094012278, −0.73905106640712407794823579834, −0.71436147937900456493159645832, −0.41182241844867992783655492954, 0.41182241844867992783655492954, 0.71436147937900456493159645832, 0.73905106640712407794823579834, 0.944765638933769613081094012278, 1.02051229791062010534719239452, 1.03685524306097078000162772402, 1.32562194773685813556297647417, 1.33090415261922327390826413942, 1.54165459308531058857531502027, 1.63035730219212311065457509288, 1.77570966479484114016352137639, 1.83525396397204908301591983170, 1.89209099181309468516671159340, 2.27980395868283393910415239732, 2.32472013418674254502329423353, 2.33123621712513767621339228602, 2.75176517843696163225953596956, 2.82664784027621376876732786567, 2.88864972284101754209685018802, 2.93306152307893602355891220264, 2.93682120733680784153308237835, 2.93768474949354172817029935327, 2.95609300377199411925271132969, 3.00095527361905711777154652304, 3.33447439434881564170680588041
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.