Dirichlet series
| L(s) = 1 | − 3·2-s + 5-s + 11·7-s + 8·8-s − 3·10-s − 15·11-s + 7·13-s − 33·14-s − 6·16-s + 3·17-s + 10·19-s + 45·22-s − 20·23-s − 20·25-s − 21·26-s − 2·29-s + 10·31-s − 7·32-s − 9·34-s + 11·35-s + 30·37-s − 30·38-s + 8·40-s + 28·41-s + 48·43-s + 60·46-s − 13·47-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.447·5-s + 4.15·7-s + 2.82·8-s − 0.948·10-s − 4.52·11-s + 1.94·13-s − 8.81·14-s − 3/2·16-s + 0.727·17-s + 2.29·19-s + 9.59·22-s − 4.17·23-s − 4·25-s − 4.11·26-s − 0.371·29-s + 1.79·31-s − 1.23·32-s − 1.54·34-s + 1.85·35-s + 4.93·37-s − 4.86·38-s + 1.26·40-s + 4.37·41-s + 7.31·43-s + 8.84·46-s − 1.89·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 239^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 239^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 239^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.59189\times 10^{14}\) |
| Root analytic conductor: | \(4.14437\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 239^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.839179185\) |
| \(L(\frac12)\) | \(\approx\) | \(9.839179185\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 239 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 + 3 T + 9 T^{2} + 19 T^{3} + 39 T^{4} + 35 p T^{5} + 63 p T^{6} + 205 T^{7} + 335 T^{8} + 529 T^{9} + 817 T^{10} + 613 p T^{11} + 1773 T^{12} + 613 p^{2} T^{13} + 817 p^{2} T^{14} + 529 p^{3} T^{15} + 335 p^{4} T^{16} + 205 p^{5} T^{17} + 63 p^{7} T^{18} + 35 p^{8} T^{19} + 39 p^{8} T^{20} + 19 p^{9} T^{21} + 9 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - T + 21 T^{2} - 24 T^{3} + 246 T^{4} - 319 T^{5} + 2218 T^{6} - 2961 T^{7} + 16728 T^{8} - 21384 T^{9} + 4221 p^{2} T^{10} - 127071 T^{11} + 113474 p T^{12} - 127071 p T^{13} + 4221 p^{4} T^{14} - 21384 p^{3} T^{15} + 16728 p^{4} T^{16} - 2961 p^{5} T^{17} + 2218 p^{6} T^{18} - 319 p^{7} T^{19} + 246 p^{8} T^{20} - 24 p^{9} T^{21} + 21 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 11 T + 93 T^{2} - 554 T^{3} + 2915 T^{4} - 13000 T^{5} + 54027 T^{6} - 202131 T^{7} + 716579 T^{8} - 2331488 T^{9} + 1031272 p T^{10} - 2958530 p T^{11} + 56795474 T^{12} - 2958530 p^{2} T^{13} + 1031272 p^{3} T^{14} - 2331488 p^{3} T^{15} + 716579 p^{4} T^{16} - 202131 p^{5} T^{17} + 54027 p^{6} T^{18} - 13000 p^{7} T^{19} + 2915 p^{8} T^{20} - 554 p^{9} T^{21} + 93 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 15 T + 16 p T^{2} + 1515 T^{3} + 11324 T^{4} + 72200 T^{5} + 415488 T^{6} + 2139486 T^{7} + 10129244 T^{8} + 43806715 T^{9} + 175845352 T^{10} + 650827169 T^{11} + 2244891678 T^{12} + 650827169 p T^{13} + 175845352 p^{2} T^{14} + 43806715 p^{3} T^{15} + 10129244 p^{4} T^{16} + 2139486 p^{5} T^{17} + 415488 p^{6} T^{18} + 72200 p^{7} T^{19} + 11324 p^{8} T^{20} + 1515 p^{9} T^{21} + 16 p^{11} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 7 T + 87 T^{2} - 346 T^{3} + 2299 T^{4} - 2836 T^{5} + 15063 T^{6} + 101811 T^{7} - 68733 T^{8} + 1560512 T^{9} + 5574538 T^{10} - 8442562 T^{11} + 157985730 T^{12} - 8442562 p T^{13} + 5574538 p^{2} T^{14} + 1560512 p^{3} T^{15} - 68733 p^{4} T^{16} + 101811 p^{5} T^{17} + 15063 p^{6} T^{18} - 2836 p^{7} T^{19} + 2299 p^{8} T^{20} - 346 p^{9} T^{21} + 87 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 3 T + 113 T^{2} - 298 T^{3} + 6742 T^{4} - 16563 T^{5} + 273746 T^{6} - 36797 p T^{7} + 8298056 T^{8} - 17581574 T^{9} + 11579393 p T^{10} - 381133325 T^{11} + 3729976282 T^{12} - 381133325 p T^{13} + 11579393 p^{3} T^{14} - 17581574 p^{3} T^{15} + 8298056 p^{4} T^{16} - 36797 p^{6} T^{17} + 273746 p^{6} T^{18} - 16563 p^{7} T^{19} + 6742 p^{8} T^{20} - 298 p^{9} T^{21} + 113 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 10 T + 160 T^{2} - 1161 T^{3} + 11463 T^{4} - 68173 T^{5} + 524846 T^{6} - 2700588 T^{7} + 17646747 T^{8} - 80812162 T^{9} + 464821650 T^{10} - 1913047944 T^{11} + 9828061466 T^{12} - 1913047944 p T^{13} + 464821650 p^{2} T^{14} - 80812162 p^{3} T^{15} + 17646747 p^{4} T^{16} - 2700588 p^{5} T^{17} + 524846 p^{6} T^{18} - 68173 p^{7} T^{19} + 11463 p^{8} T^{20} - 1161 p^{9} T^{21} + 160 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 20 T + 348 T^{2} + 4267 T^{3} + 46241 T^{4} + 423251 T^{5} + 3511868 T^{6} + 25995766 T^{7} + 177347091 T^{8} + 1104234322 T^{9} + 6395549608 T^{10} + 34150914992 T^{11} + 170322702998 T^{12} + 34150914992 p T^{13} + 6395549608 p^{2} T^{14} + 1104234322 p^{3} T^{15} + 177347091 p^{4} T^{16} + 25995766 p^{5} T^{17} + 3511868 p^{6} T^{18} + 423251 p^{7} T^{19} + 46241 p^{8} T^{20} + 4267 p^{9} T^{21} + 348 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 2 T + 155 T^{2} + 162 T^{3} + 12302 T^{4} + 3604 T^{5} + 678598 T^{6} - 162720 T^{7} + 29679460 T^{8} - 14415978 T^{9} + 1094315211 T^{10} - 578142430 T^{11} + 34389303026 T^{12} - 578142430 p T^{13} + 1094315211 p^{2} T^{14} - 14415978 p^{3} T^{15} + 29679460 p^{4} T^{16} - 162720 p^{5} T^{17} + 678598 p^{6} T^{18} + 3604 p^{7} T^{19} + 12302 p^{8} T^{20} + 162 p^{9} T^{21} + 155 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 10 T + 130 T^{2} - 1229 T^{3} + 12132 T^{4} - 96059 T^{5} + 773500 T^{6} - 5455669 T^{7} + 38720748 T^{8} - 246037425 T^{9} + 1572968622 T^{10} - 9126930404 T^{11} + 53072451462 T^{12} - 9126930404 p T^{13} + 1572968622 p^{2} T^{14} - 246037425 p^{3} T^{15} + 38720748 p^{4} T^{16} - 5455669 p^{5} T^{17} + 773500 p^{6} T^{18} - 96059 p^{7} T^{19} + 12132 p^{8} T^{20} - 1229 p^{9} T^{21} + 130 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 30 T + 466 T^{2} - 5015 T^{3} + 1219 p T^{4} - 382857 T^{5} + 3136742 T^{6} - 24085180 T^{7} + 173860227 T^{8} - 1211654662 T^{9} + 8225770648 T^{10} - 53676423524 T^{11} + 334329993818 T^{12} - 53676423524 p T^{13} + 8225770648 p^{2} T^{14} - 1211654662 p^{3} T^{15} + 173860227 p^{4} T^{16} - 24085180 p^{5} T^{17} + 3136742 p^{6} T^{18} - 382857 p^{7} T^{19} + 1219 p^{9} T^{20} - 5015 p^{9} T^{21} + 466 p^{10} T^{22} - 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 28 T + 625 T^{2} - 9904 T^{3} + 138547 T^{4} - 1634562 T^{5} + 17635471 T^{6} - 169521842 T^{7} + 1516174411 T^{8} - 12376446530 T^{9} + 94816719696 T^{10} - 670059249982 T^{11} + 4460214760194 T^{12} - 670059249982 p T^{13} + 94816719696 p^{2} T^{14} - 12376446530 p^{3} T^{15} + 1516174411 p^{4} T^{16} - 169521842 p^{5} T^{17} + 17635471 p^{6} T^{18} - 1634562 p^{7} T^{19} + 138547 p^{8} T^{20} - 9904 p^{9} T^{21} + 625 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 48 T + 1326 T^{2} - 26581 T^{3} + 428219 T^{4} - 5833557 T^{5} + 69372908 T^{6} - 734828058 T^{7} + 7030086507 T^{8} - 61323149970 T^{9} + 490950661110 T^{10} - 3622972496268 T^{11} + 24704850494738 T^{12} - 3622972496268 p T^{13} + 490950661110 p^{2} T^{14} - 61323149970 p^{3} T^{15} + 7030086507 p^{4} T^{16} - 734828058 p^{5} T^{17} + 69372908 p^{6} T^{18} - 5833557 p^{7} T^{19} + 428219 p^{8} T^{20} - 26581 p^{9} T^{21} + 1326 p^{10} T^{22} - 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 13 T + 361 T^{2} + 2972 T^{3} + 53319 T^{4} + 325660 T^{5} + 5201693 T^{6} + 26159887 T^{7} + 397799595 T^{8} + 1704466282 T^{9} + 24584470554 T^{10} + 92273200312 T^{11} + 1259253382938 T^{12} + 92273200312 p T^{13} + 24584470554 p^{2} T^{14} + 1704466282 p^{3} T^{15} + 397799595 p^{4} T^{16} + 26159887 p^{5} T^{17} + 5201693 p^{6} T^{18} + 325660 p^{7} T^{19} + 53319 p^{8} T^{20} + 2972 p^{9} T^{21} + 361 p^{10} T^{22} + 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 2 T + 258 T^{2} - 93 T^{3} + 32885 T^{4} + 43439 T^{5} + 2877824 T^{6} + 7693030 T^{7} + 207314035 T^{8} + 671437404 T^{9} + 13347968798 T^{10} + 41771034610 T^{11} + 760230981102 T^{12} + 41771034610 p T^{13} + 13347968798 p^{2} T^{14} + 671437404 p^{3} T^{15} + 207314035 p^{4} T^{16} + 7693030 p^{5} T^{17} + 2877824 p^{6} T^{18} + 43439 p^{7} T^{19} + 32885 p^{8} T^{20} - 93 p^{9} T^{21} + 258 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 14 T + 379 T^{2} - 5262 T^{3} + 83959 T^{4} - 1031008 T^{5} + 12639879 T^{6} - 136319784 T^{7} + 1407179659 T^{8} - 13306131298 T^{9} + 119973704198 T^{10} - 1001392116634 T^{11} + 7988641850554 T^{12} - 1001392116634 p T^{13} + 119973704198 p^{2} T^{14} - 13306131298 p^{3} T^{15} + 1407179659 p^{4} T^{16} - 136319784 p^{5} T^{17} + 12639879 p^{6} T^{18} - 1031008 p^{7} T^{19} + 83959 p^{8} T^{20} - 5262 p^{9} T^{21} + 379 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 14 T + 294 T^{2} - 3285 T^{3} + 47990 T^{4} - 479161 T^{5} + 5808988 T^{6} - 52690929 T^{7} + 561371086 T^{8} - 4638208751 T^{9} + 44299014430 T^{10} - 337767900860 T^{11} + 2944154930550 T^{12} - 337767900860 p T^{13} + 44299014430 p^{2} T^{14} - 4638208751 p^{3} T^{15} + 561371086 p^{4} T^{16} - 52690929 p^{5} T^{17} + 5808988 p^{6} T^{18} - 479161 p^{7} T^{19} + 47990 p^{8} T^{20} - 3285 p^{9} T^{21} + 294 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 52 T + 1842 T^{2} - 46748 T^{3} + 977943 T^{4} - 17122352 T^{5} + 262901634 T^{6} - 3562988448 T^{7} + 43592254395 T^{8} - 482396186796 T^{9} + 4889437880732 T^{10} - 45333619130132 T^{11} + 387334370872986 T^{12} - 45333619130132 p T^{13} + 4889437880732 p^{2} T^{14} - 482396186796 p^{3} T^{15} + 43592254395 p^{4} T^{16} - 3562988448 p^{5} T^{17} + 262901634 p^{6} T^{18} - 17122352 p^{7} T^{19} + 977943 p^{8} T^{20} - 46748 p^{9} T^{21} + 1842 p^{10} T^{22} - 52 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 7 T + 517 T^{2} - 2564 T^{3} + 126703 T^{4} - 451596 T^{5} + 20501341 T^{6} - 54857821 T^{7} + 2502987379 T^{8} - 5363320182 T^{9} + 242994722958 T^{10} - 443860473360 T^{11} + 19115398243386 T^{12} - 443860473360 p T^{13} + 242994722958 p^{2} T^{14} - 5363320182 p^{3} T^{15} + 2502987379 p^{4} T^{16} - 54857821 p^{5} T^{17} + 20501341 p^{6} T^{18} - 451596 p^{7} T^{19} + 126703 p^{8} T^{20} - 2564 p^{9} T^{21} + 517 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 14 T + 470 T^{2} - 3921 T^{3} + 82995 T^{4} - 351471 T^{5} + 8171174 T^{6} - 5576012 T^{7} + 706739371 T^{8} + 225643690 T^{9} + 74008396324 T^{10} - 79127959784 T^{11} + 6563524616002 T^{12} - 79127959784 p T^{13} + 74008396324 p^{2} T^{14} + 225643690 p^{3} T^{15} + 706739371 p^{4} T^{16} - 5576012 p^{5} T^{17} + 8171174 p^{6} T^{18} - 351471 p^{7} T^{19} + 82995 p^{8} T^{20} - 3921 p^{9} T^{21} + 470 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 15 T + 718 T^{2} - 8877 T^{3} + 232723 T^{4} - 2402693 T^{5} + 45651448 T^{6} - 398354235 T^{7} + 78437909 p T^{8} - 46614315644 T^{9} + 641429776394 T^{10} - 4305099672584 T^{11} + 54819251065090 T^{12} - 4305099672584 p T^{13} + 641429776394 p^{2} T^{14} - 46614315644 p^{3} T^{15} + 78437909 p^{5} T^{16} - 398354235 p^{5} T^{17} + 45651448 p^{6} T^{18} - 2402693 p^{7} T^{19} + 232723 p^{8} T^{20} - 8877 p^{9} T^{21} + 718 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 29 T + 1026 T^{2} + 22407 T^{3} + 472618 T^{4} + 8171314 T^{5} + 130547428 T^{6} + 1851659452 T^{7} + 24246546242 T^{8} + 289059243015 T^{9} + 3204042262322 T^{10} + 32586909035499 T^{11} + 309662575482134 T^{12} + 32586909035499 p T^{13} + 3204042262322 p^{2} T^{14} + 289059243015 p^{3} T^{15} + 24246546242 p^{4} T^{16} + 1851659452 p^{5} T^{17} + 130547428 p^{6} T^{18} + 8171314 p^{7} T^{19} + 472618 p^{8} T^{20} + 22407 p^{9} T^{21} + 1026 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 71 T + 2926 T^{2} - 86373 T^{3} + 2030049 T^{4} - 39930313 T^{5} + 680968380 T^{6} - 10291017263 T^{7} + 140199554899 T^{8} - 1741241331882 T^{9} + 19888385148006 T^{10} - 209973161307586 T^{11} + 2056150235605430 T^{12} - 209973161307586 p T^{13} + 19888385148006 p^{2} T^{14} - 1741241331882 p^{3} T^{15} + 140199554899 p^{4} T^{16} - 10291017263 p^{5} T^{17} + 680968380 p^{6} T^{18} - 39930313 p^{7} T^{19} + 2030049 p^{8} T^{20} - 86373 p^{9} T^{21} + 2926 p^{10} T^{22} - 71 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 515 T^{2} - 644 T^{3} + 137421 T^{4} - 190722 T^{5} + 24933023 T^{6} - 24913490 T^{7} + 3449635427 T^{8} - 1085922902 T^{9} + 397469856734 T^{10} + 83683389486 T^{11} + 40444052447518 T^{12} + 83683389486 p T^{13} + 397469856734 p^{2} T^{14} - 1085922902 p^{3} T^{15} + 3449635427 p^{4} T^{16} - 24913490 p^{5} T^{17} + 24933023 p^{6} T^{18} - 190722 p^{7} T^{19} + 137421 p^{8} T^{20} - 644 p^{9} T^{21} + 515 p^{10} T^{22} + p^{12} T^{24} \) | |
| 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
−2.63341393039785032305735463172, −2.55618893961075310798613150260, −2.50954484424773842606958206760, −2.42761630640656466772952145069, −2.40935182101588641102256319816, −2.36572918391735230738861633985, −2.22122655795198965792880100700, −2.02721097196878560972268536282, −2.01188395515728202827994106607, −1.88326324805813234694592133263, −1.87758229163159449719252200415, −1.86321240347838209672644752716, −1.81913564516356011570682489816, −1.52618191939321463959500906188, −1.50933392616090859239687171426, −1.14289822221131708816077949759, −1.00069469978187152852826015441, −0.912610241298873622151152542190, −0.885751021362649894374156425018, −0.77782400340100948676482026473, −0.74777494570611780662537224233, −0.70143497148084560055447732391, −0.48803269594860323437092360751, −0.43518572569401739192468730358, −0.25580410885524105095536908903, 0.25580410885524105095536908903, 0.43518572569401739192468730358, 0.48803269594860323437092360751, 0.70143497148084560055447732391, 0.74777494570611780662537224233, 0.77782400340100948676482026473, 0.885751021362649894374156425018, 0.912610241298873622151152542190, 1.00069469978187152852826015441, 1.14289822221131708816077949759, 1.50933392616090859239687171426, 1.52618191939321463959500906188, 1.81913564516356011570682489816, 1.86321240347838209672644752716, 1.87758229163159449719252200415, 1.88326324805813234694592133263, 2.01188395515728202827994106607, 2.02721097196878560972268536282, 2.22122655795198965792880100700, 2.36572918391735230738861633985, 2.40935182101588641102256319816, 2.42761630640656466772952145069, 2.50954484424773842606958206760, 2.55618893961075310798613150260, 2.63341393039785032305735463172
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.