Dirichlet series
| L(s) = 1 | − 5·3-s + 5·5-s − 8·7-s + 2·9-s − 18·11-s + 4·13-s − 25·15-s − 2·17-s − 6·19-s + 40·21-s − 13·23-s − 17·25-s + 34·27-s + 20·29-s − 7·31-s + 90·33-s − 40·35-s + 10·37-s − 20·39-s + 2·41-s − 8·43-s + 10·45-s − 12·47-s − 8·49-s + 10·51-s + 12·53-s − 90·55-s + ⋯ |
| L(s) = 1 | − 2.88·3-s + 2.23·5-s − 3.02·7-s + 2/3·9-s − 5.42·11-s + 1.10·13-s − 6.45·15-s − 0.485·17-s − 1.37·19-s + 8.72·21-s − 2.71·23-s − 3.39·25-s + 6.54·27-s + 3.71·29-s − 1.25·31-s + 15.6·33-s − 6.76·35-s + 1.64·37-s − 3.20·39-s + 0.312·41-s − 1.21·43-s + 1.49·45-s − 1.75·47-s − 8/7·49-s + 1.40·51-s + 1.64·53-s − 12.1·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 503^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 503^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 503^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.96113\times 10^{21}\) |
| Root analytic conductor: | \(8.01645\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{48} \cdot 503^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 503 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 + 5 T + 23 T^{2} + 71 T^{3} + 73 p T^{4} + 553 T^{5} + 1403 T^{6} + 3112 T^{7} + 6916 T^{8} + 13793 T^{9} + 3046 p^{2} T^{10} + 5516 p^{2} T^{11} + 89792 T^{12} + 5516 p^{3} T^{13} + 3046 p^{4} T^{14} + 13793 p^{3} T^{15} + 6916 p^{4} T^{16} + 3112 p^{5} T^{17} + 1403 p^{6} T^{18} + 553 p^{7} T^{19} + 73 p^{9} T^{20} + 71 p^{9} T^{21} + 23 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - p T + 42 T^{2} - 154 T^{3} + 791 T^{4} - 2412 T^{5} + 9636 T^{6} - 25656 T^{7} + 86256 T^{8} - 204402 T^{9} + 600308 T^{10} - 51099 p^{2} T^{11} + 3342092 T^{12} - 51099 p^{3} T^{13} + 600308 p^{2} T^{14} - 204402 p^{3} T^{15} + 86256 p^{4} T^{16} - 25656 p^{5} T^{17} + 9636 p^{6} T^{18} - 2412 p^{7} T^{19} + 791 p^{8} T^{20} - 154 p^{9} T^{21} + 42 p^{10} T^{22} - p^{12} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 8 T + 72 T^{2} + 57 p T^{3} + 2199 T^{4} + 9669 T^{5} + 41150 T^{6} + 152258 T^{7} + 542867 T^{8} + 1744965 T^{9} + 774505 p T^{10} + 15427129 T^{11} + 42558007 T^{12} + 15427129 p T^{13} + 774505 p^{3} T^{14} + 1744965 p^{3} T^{15} + 542867 p^{4} T^{16} + 152258 p^{5} T^{17} + 41150 p^{6} T^{18} + 9669 p^{7} T^{19} + 2199 p^{8} T^{20} + 57 p^{10} T^{21} + 72 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 18 T + 236 T^{2} + 2219 T^{3} + 17491 T^{4} + 115780 T^{5} + 677529 T^{6} + 3514943 T^{7} + 16587530 T^{8} + 71330085 T^{9} + 283931035 T^{10} + 95050927 p T^{11} + 3596181460 T^{12} + 95050927 p^{2} T^{13} + 283931035 p^{2} T^{14} + 71330085 p^{3} T^{15} + 16587530 p^{4} T^{16} + 3514943 p^{5} T^{17} + 677529 p^{6} T^{18} + 115780 p^{7} T^{19} + 17491 p^{8} T^{20} + 2219 p^{9} T^{21} + 236 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 4 T + 95 T^{2} - 278 T^{3} + 4083 T^{4} - 8607 T^{5} + 110337 T^{6} - 159728 T^{7} + 168114 p T^{8} - 2023466 T^{9} + 34878800 T^{10} - 21192537 T^{11} + 480640372 T^{12} - 21192537 p T^{13} + 34878800 p^{2} T^{14} - 2023466 p^{3} T^{15} + 168114 p^{5} T^{16} - 159728 p^{5} T^{17} + 110337 p^{6} T^{18} - 8607 p^{7} T^{19} + 4083 p^{8} T^{20} - 278 p^{9} T^{21} + 95 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 2 T + 86 T^{2} + 211 T^{3} + 3520 T^{4} + 11022 T^{5} + 97722 T^{6} + 347875 T^{7} + 2230039 T^{8} + 7425255 T^{9} + 46049968 T^{10} + 7516115 p T^{11} + 844910160 T^{12} + 7516115 p^{2} T^{13} + 46049968 p^{2} T^{14} + 7425255 p^{3} T^{15} + 2230039 p^{4} T^{16} + 347875 p^{5} T^{17} + 97722 p^{6} T^{18} + 11022 p^{7} T^{19} + 3520 p^{8} T^{20} + 211 p^{9} T^{21} + 86 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 6 T + 179 T^{2} + 1019 T^{3} + 15567 T^{4} + 81493 T^{5} + 860771 T^{6} + 4072782 T^{7} + 33490409 T^{8} + 141658777 T^{9} + 962981822 T^{10} + 3604155843 T^{11} + 20956227758 T^{12} + 3604155843 p T^{13} + 962981822 p^{2} T^{14} + 141658777 p^{3} T^{15} + 33490409 p^{4} T^{16} + 4072782 p^{5} T^{17} + 860771 p^{6} T^{18} + 81493 p^{7} T^{19} + 15567 p^{8} T^{20} + 1019 p^{9} T^{21} + 179 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 13 T + 237 T^{2} + 2002 T^{3} + 21372 T^{4} + 133225 T^{5} + 1071017 T^{6} + 5181303 T^{7} + 35330634 T^{8} + 137041097 T^{9} + 885747404 T^{10} + 2981518951 T^{11} + 20222903567 T^{12} + 2981518951 p T^{13} + 885747404 p^{2} T^{14} + 137041097 p^{3} T^{15} + 35330634 p^{4} T^{16} + 5181303 p^{5} T^{17} + 1071017 p^{6} T^{18} + 133225 p^{7} T^{19} + 21372 p^{8} T^{20} + 2002 p^{9} T^{21} + 237 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 20 T + 399 T^{2} - 4927 T^{3} + 59113 T^{4} - 549200 T^{5} + 4994143 T^{6} - 38035759 T^{7} + 285978915 T^{8} - 1866713053 T^{9} + 12107702506 T^{10} - 69364622861 T^{11} + 396392131942 T^{12} - 69364622861 p T^{13} + 12107702506 p^{2} T^{14} - 1866713053 p^{3} T^{15} + 285978915 p^{4} T^{16} - 38035759 p^{5} T^{17} + 4994143 p^{6} T^{18} - 549200 p^{7} T^{19} + 59113 p^{8} T^{20} - 4927 p^{9} T^{21} + 399 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 7 T + 282 T^{2} + 1830 T^{3} + 38139 T^{4} + 230297 T^{5} + 3280516 T^{6} + 18364168 T^{7} + 200366767 T^{8} + 1028922293 T^{9} + 9171046402 T^{10} + 42467891207 T^{11} + 322887365354 T^{12} + 42467891207 p T^{13} + 9171046402 p^{2} T^{14} + 1028922293 p^{3} T^{15} + 200366767 p^{4} T^{16} + 18364168 p^{5} T^{17} + 3280516 p^{6} T^{18} + 230297 p^{7} T^{19} + 38139 p^{8} T^{20} + 1830 p^{9} T^{21} + 282 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 10 T + 243 T^{2} - 1891 T^{3} + 28373 T^{4} - 204733 T^{5} + 2343397 T^{6} - 15984122 T^{7} + 146808103 T^{8} - 936951483 T^{9} + 7321030232 T^{10} - 43550538569 T^{11} + 299802706494 T^{12} - 43550538569 p T^{13} + 7321030232 p^{2} T^{14} - 936951483 p^{3} T^{15} + 146808103 p^{4} T^{16} - 15984122 p^{5} T^{17} + 2343397 p^{6} T^{18} - 204733 p^{7} T^{19} + 28373 p^{8} T^{20} - 1891 p^{9} T^{21} + 243 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 2 T + 310 T^{2} - 785 T^{3} + 47098 T^{4} - 138456 T^{5} + 4673492 T^{6} - 14922727 T^{7} + 340248619 T^{8} - 1111165915 T^{9} + 19293798486 T^{10} - 60634376507 T^{11} + 878545992372 T^{12} - 60634376507 p T^{13} + 19293798486 p^{2} T^{14} - 1111165915 p^{3} T^{15} + 340248619 p^{4} T^{16} - 14922727 p^{5} T^{17} + 4673492 p^{6} T^{18} - 138456 p^{7} T^{19} + 47098 p^{8} T^{20} - 785 p^{9} T^{21} + 310 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 8 T + 333 T^{2} + 1866 T^{3} + 49961 T^{4} + 4677 p T^{5} + 4798493 T^{6} + 14405750 T^{7} + 347174076 T^{8} + 838886304 T^{9} + 20293592746 T^{10} + 42545902371 T^{11} + 968100841052 T^{12} + 42545902371 p T^{13} + 20293592746 p^{2} T^{14} + 838886304 p^{3} T^{15} + 347174076 p^{4} T^{16} + 14405750 p^{5} T^{17} + 4798493 p^{6} T^{18} + 4677 p^{8} T^{19} + 49961 p^{8} T^{20} + 1866 p^{9} T^{21} + 333 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 12 T + 302 T^{2} + 3187 T^{3} + 47279 T^{4} + 415791 T^{5} + 4717830 T^{6} + 35652736 T^{7} + 339864289 T^{8} + 2268004051 T^{9} + 19491144185 T^{10} + 119470683497 T^{11} + 965417437421 T^{12} + 119470683497 p T^{13} + 19491144185 p^{2} T^{14} + 2268004051 p^{3} T^{15} + 339864289 p^{4} T^{16} + 35652736 p^{5} T^{17} + 4717830 p^{6} T^{18} + 415791 p^{7} T^{19} + 47279 p^{8} T^{20} + 3187 p^{9} T^{21} + 302 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 12 T + 545 T^{2} - 5561 T^{3} + 137317 T^{4} - 1216285 T^{5} + 21437635 T^{6} - 166788154 T^{7} + 2329758205 T^{8} - 15995580255 T^{9} + 186586431716 T^{10} - 1128590015573 T^{11} + 11317232997794 T^{12} - 1128590015573 p T^{13} + 186586431716 p^{2} T^{14} - 15995580255 p^{3} T^{15} + 2329758205 p^{4} T^{16} - 166788154 p^{5} T^{17} + 21437635 p^{6} T^{18} - 1216285 p^{7} T^{19} + 137317 p^{8} T^{20} - 5561 p^{9} T^{21} + 545 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 6 T + 277 T^{2} + 1124 T^{3} + 34762 T^{4} + 84929 T^{5} + 3002529 T^{6} + 4076375 T^{7} + 229711951 T^{8} + 229401713 T^{9} + 16358116930 T^{10} + 15943411277 T^{11} + 1033672281260 T^{12} + 15943411277 p T^{13} + 16358116930 p^{2} T^{14} + 229401713 p^{3} T^{15} + 229711951 p^{4} T^{16} + 4076375 p^{5} T^{17} + 3002529 p^{6} T^{18} + 84929 p^{7} T^{19} + 34762 p^{8} T^{20} + 1124 p^{9} T^{21} + 277 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 10 T + 365 T^{2} + 3179 T^{3} + 61581 T^{4} + 449878 T^{5} + 6491041 T^{6} + 40615545 T^{7} + 512777084 T^{8} + 2961728657 T^{9} + 35403589926 T^{10} + 197962455527 T^{11} + 2258050315556 T^{12} + 197962455527 p T^{13} + 35403589926 p^{2} T^{14} + 2961728657 p^{3} T^{15} + 512777084 p^{4} T^{16} + 40615545 p^{5} T^{17} + 6491041 p^{6} T^{18} + 449878 p^{7} T^{19} + 61581 p^{8} T^{20} + 3179 p^{9} T^{21} + 365 p^{10} T^{22} + 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 7 T + 554 T^{2} + 3317 T^{3} + 145357 T^{4} + 743955 T^{5} + 24141915 T^{6} + 105743608 T^{7} + 2874222370 T^{8} + 10870454891 T^{9} + 263904792235 T^{10} + 880117558364 T^{11} + 19537385096368 T^{12} + 880117558364 p T^{13} + 263904792235 p^{2} T^{14} + 10870454891 p^{3} T^{15} + 2874222370 p^{4} T^{16} + 105743608 p^{5} T^{17} + 24141915 p^{6} T^{18} + 743955 p^{7} T^{19} + 145357 p^{8} T^{20} + 3317 p^{9} T^{21} + 554 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 22 T + 767 T^{2} + 12443 T^{3} + 245151 T^{4} + 3126102 T^{5} + 44600111 T^{6} + 466232143 T^{7} + 5338265595 T^{8} + 47751004997 T^{9} + 474162762402 T^{10} + 3842492624367 T^{11} + 35488548426314 T^{12} + 3842492624367 p T^{13} + 474162762402 p^{2} T^{14} + 47751004997 p^{3} T^{15} + 5338265595 p^{4} T^{16} + 466232143 p^{5} T^{17} + 44600111 p^{6} T^{18} + 3126102 p^{7} T^{19} + 245151 p^{8} T^{20} + 12443 p^{9} T^{21} + 767 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 23 T + 914 T^{2} + 15862 T^{3} + 359607 T^{4} + 5033434 T^{5} + 83581420 T^{6} + 980185074 T^{7} + 13078758902 T^{8} + 131435921598 T^{9} + 1474371678576 T^{10} + 12844486325361 T^{11} + 123956533139464 T^{12} + 12844486325361 p T^{13} + 1474371678576 p^{2} T^{14} + 131435921598 p^{3} T^{15} + 13078758902 p^{4} T^{16} + 980185074 p^{5} T^{17} + 83581420 p^{6} T^{18} + 5033434 p^{7} T^{19} + 359607 p^{8} T^{20} + 15862 p^{9} T^{21} + 914 p^{10} T^{22} + 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 13 T + 740 T^{2} + 9277 T^{3} + 264468 T^{4} + 3122234 T^{5} + 60133963 T^{6} + 655134600 T^{7} + 9654177811 T^{8} + 95352923720 T^{9} + 1149494991777 T^{10} + 10118852614358 T^{11} + 103879296399120 T^{12} + 10118852614358 p T^{13} + 1149494991777 p^{2} T^{14} + 95352923720 p^{3} T^{15} + 9654177811 p^{4} T^{16} + 655134600 p^{5} T^{17} + 60133963 p^{6} T^{18} + 3122234 p^{7} T^{19} + 264468 p^{8} T^{20} + 9277 p^{9} T^{21} + 740 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + T + 591 T^{2} - 693 T^{3} + 167307 T^{4} - 517795 T^{5} + 30927255 T^{6} - 144179528 T^{7} + 4250890052 T^{8} - 291958985 p T^{9} + 465398797158 T^{10} - 2800671733828 T^{11} + 42176704430232 T^{12} - 2800671733828 p T^{13} + 465398797158 p^{2} T^{14} - 291958985 p^{4} T^{15} + 4250890052 p^{4} T^{16} - 144179528 p^{5} T^{17} + 30927255 p^{6} T^{18} - 517795 p^{7} T^{19} + 167307 p^{8} T^{20} - 693 p^{9} T^{21} + 591 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 3 T + 559 T^{2} + 2334 T^{3} + 165630 T^{4} + 762205 T^{5} + 33875739 T^{6} + 156779452 T^{7} + 5245514383 T^{8} + 23317169957 T^{9} + 641955466630 T^{10} + 2655079688665 T^{11} + 63398187090724 T^{12} + 2655079688665 p T^{13} + 641955466630 p^{2} T^{14} + 23317169957 p^{3} T^{15} + 5245514383 p^{4} T^{16} + 156779452 p^{5} T^{17} + 33875739 p^{6} T^{18} + 762205 p^{7} T^{19} + 165630 p^{8} T^{20} + 2334 p^{9} T^{21} + 559 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 70 T + 2935 T^{2} + 87469 T^{3} + 2060013 T^{4} + 39911519 T^{5} + 658581287 T^{6} + 9431186506 T^{7} + 119768221401 T^{8} + 1373580747687 T^{9} + 14608461788988 T^{10} + 147941064750717 T^{11} + 1466383547078142 T^{12} + 147941064750717 p T^{13} + 14608461788988 p^{2} T^{14} + 1373580747687 p^{3} T^{15} + 119768221401 p^{4} T^{16} + 9431186506 p^{5} T^{17} + 658581287 p^{6} T^{18} + 39911519 p^{7} T^{19} + 2060013 p^{8} T^{20} + 87469 p^{9} T^{21} + 2935 p^{10} T^{22} + 70 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.81872587132273295067914161849, −2.54132689184573335576675520980, −2.47813573559043117911835710964, −2.46843210895920392019481007562, −2.41045066385317163337635020398, −2.40912147543860839901780492988, −2.34875642004092433803056987412, −2.25134353381470241859767809581, −2.20738007057015899677536215681, −2.14185257590308808408001765561, −2.05775738733835497052874644604, −1.92367986882436512274049697936, −1.80901763556065051963975050934, −1.71047941431144658563639608666, −1.68395328218789227082989817477, −1.54705744694947460437669292511, −1.52070518294249384762898962920, −1.30583960572013904738199398748, −1.20218043659761289514704718627, −1.18142570961746241554222612551, −1.11338766397540841458771282534, −1.09217407257845442424265398076, −1.06603667989526294435090719764, −0.843589061980310468372746125277, −0.64757576686680652641199035699, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.64757576686680652641199035699, 0.843589061980310468372746125277, 1.06603667989526294435090719764, 1.09217407257845442424265398076, 1.11338766397540841458771282534, 1.18142570961746241554222612551, 1.20218043659761289514704718627, 1.30583960572013904738199398748, 1.52070518294249384762898962920, 1.54705744694947460437669292511, 1.68395328218789227082989817477, 1.71047941431144658563639608666, 1.80901763556065051963975050934, 1.92367986882436512274049697936, 2.05775738733835497052874644604, 2.14185257590308808408001765561, 2.20738007057015899677536215681, 2.25134353381470241859767809581, 2.34875642004092433803056987412, 2.40912147543860839901780492988, 2.41045066385317163337635020398, 2.46843210895920392019481007562, 2.47813573559043117911835710964, 2.54132689184573335576675520980, 2.81872587132273295067914161849
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.