Dirichlet series
| L(s) = 1 | − 3-s + 6·5-s − 3·7-s − 10·9-s − 22·11-s − 5·13-s − 6·15-s − 4·17-s + 6·19-s + 3·21-s − 32·23-s − 10·25-s + 13·27-s + 6·29-s − 8·31-s + 22·33-s − 18·35-s − 8·37-s + 5·39-s − 41-s + 2·43-s − 60·45-s − 34·47-s − 42·49-s + 4·51-s + 5·53-s − 132·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 2.68·5-s − 1.13·7-s − 3.33·9-s − 6.63·11-s − 1.38·13-s − 1.54·15-s − 0.970·17-s + 1.37·19-s + 0.654·21-s − 6.67·23-s − 2·25-s + 2.50·27-s + 1.11·29-s − 1.43·31-s + 3.82·33-s − 3.04·35-s − 1.31·37-s + 0.800·39-s − 0.156·41-s + 0.304·43-s − 8.94·45-s − 4.95·47-s − 6·49-s + 0.560·51-s + 0.686·53-s − 17.7·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 241^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 241^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 241^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.26055\times 10^{17}\) |
| Root analytic conductor: | \(5.54890\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(12\) |
| Selberg data: | \((24,\ 2^{48} \cdot 241^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
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 | \( 1 \) |
| 241 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 + T + 11 T^{2} + 8 T^{3} + 68 T^{4} + 10 p T^{5} + 101 p T^{6} + 40 T^{7} + 1099 T^{8} - 65 p T^{9} + 3502 T^{10} - 1444 T^{11} + 10528 T^{12} - 1444 p T^{13} + 3502 p^{2} T^{14} - 65 p^{4} T^{15} + 1099 p^{4} T^{16} + 40 p^{5} T^{17} + 101 p^{7} T^{18} + 10 p^{8} T^{19} + 68 p^{8} T^{20} + 8 p^{9} T^{21} + 11 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 6 T + 46 T^{2} - 196 T^{3} + 882 T^{4} - 3017 T^{5} + 10331 T^{6} - 30548 T^{7} + 88612 T^{8} - 235429 T^{9} + 605854 T^{10} - 1456907 T^{11} + 3364502 T^{12} - 1456907 p T^{13} + 605854 p^{2} T^{14} - 235429 p^{3} T^{15} + 88612 p^{4} T^{16} - 30548 p^{5} T^{17} + 10331 p^{6} T^{18} - 3017 p^{7} T^{19} + 882 p^{8} T^{20} - 196 p^{9} T^{21} + 46 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 3 T + 51 T^{2} + 135 T^{3} + 167 p T^{4} + 59 p^{2} T^{5} + 16678 T^{6} + 41432 T^{7} + 25259 p T^{8} + 460010 T^{9} + 1555215 T^{10} + 4088130 T^{11} + 11751926 T^{12} + 4088130 p T^{13} + 1555215 p^{2} T^{14} + 460010 p^{3} T^{15} + 25259 p^{5} T^{16} + 41432 p^{5} T^{17} + 16678 p^{6} T^{18} + 59 p^{9} T^{19} + 167 p^{9} T^{20} + 135 p^{9} T^{21} + 51 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 2 p T + 309 T^{2} + 3215 T^{3} + 27241 T^{4} + 195612 T^{5} + 1224926 T^{6} + 6803622 T^{7} + 33951641 T^{8} + 153501494 T^{9} + 632754137 T^{10} + 2387213463 T^{11} + 8263362466 T^{12} + 2387213463 p T^{13} + 632754137 p^{2} T^{14} + 153501494 p^{3} T^{15} + 33951641 p^{4} T^{16} + 6803622 p^{5} T^{17} + 1224926 p^{6} T^{18} + 195612 p^{7} T^{19} + 27241 p^{8} T^{20} + 3215 p^{9} T^{21} + 309 p^{10} T^{22} + 2 p^{12} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 5 T + 94 T^{2} + 419 T^{3} + 4519 T^{4} + 18313 T^{5} + 144981 T^{6} + 535786 T^{7} + 3431095 T^{8} + 11548019 T^{9} + 62910413 T^{10} + 191545534 T^{11} + 914969618 T^{12} + 191545534 p T^{13} + 62910413 p^{2} T^{14} + 11548019 p^{3} T^{15} + 3431095 p^{4} T^{16} + 535786 p^{5} T^{17} + 144981 p^{6} T^{18} + 18313 p^{7} T^{19} + 4519 p^{8} T^{20} + 419 p^{9} T^{21} + 94 p^{10} T^{22} + 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 4 T + 107 T^{2} + 378 T^{3} + 5581 T^{4} + 16942 T^{5} + 191746 T^{6} + 492741 T^{7} + 4974231 T^{8} + 10970245 T^{9} + 105662523 T^{10} + 208156762 T^{11} + 1924983462 T^{12} + 208156762 p T^{13} + 105662523 p^{2} T^{14} + 10970245 p^{3} T^{15} + 4974231 p^{4} T^{16} + 492741 p^{5} T^{17} + 191746 p^{6} T^{18} + 16942 p^{7} T^{19} + 5581 p^{8} T^{20} + 378 p^{9} T^{21} + 107 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 6 T + 142 T^{2} - 730 T^{3} + 10024 T^{4} - 2443 p T^{5} + 468739 T^{6} - 1979928 T^{7} + 16137130 T^{8} - 62321901 T^{9} + 430966450 T^{10} - 1510794001 T^{11} + 9148767508 T^{12} - 1510794001 p T^{13} + 430966450 p^{2} T^{14} - 62321901 p^{3} T^{15} + 16137130 p^{4} T^{16} - 1979928 p^{5} T^{17} + 468739 p^{6} T^{18} - 2443 p^{8} T^{19} + 10024 p^{8} T^{20} - 730 p^{9} T^{21} + 142 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 32 T + 580 T^{2} + 7469 T^{3} + 76528 T^{4} + 663240 T^{5} + 5077858 T^{6} + 35272611 T^{7} + 225993748 T^{8} + 1346655168 T^{9} + 7505092384 T^{10} + 39288214509 T^{11} + 193919253422 T^{12} + 39288214509 p T^{13} + 7505092384 p^{2} T^{14} + 1346655168 p^{3} T^{15} + 225993748 p^{4} T^{16} + 35272611 p^{5} T^{17} + 5077858 p^{6} T^{18} + 663240 p^{7} T^{19} + 76528 p^{8} T^{20} + 7469 p^{9} T^{21} + 580 p^{10} T^{22} + 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 6 T + 135 T^{2} - 539 T^{3} + 8952 T^{4} - 35377 T^{5} + 16518 p T^{6} - 1973598 T^{7} + 19968664 T^{8} - 81475880 T^{9} + 699787907 T^{10} - 2930882253 T^{11} + 22109454828 T^{12} - 2930882253 p T^{13} + 699787907 p^{2} T^{14} - 81475880 p^{3} T^{15} + 19968664 p^{4} T^{16} - 1973598 p^{5} T^{17} + 16518 p^{7} T^{18} - 35377 p^{7} T^{19} + 8952 p^{8} T^{20} - 539 p^{9} T^{21} + 135 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 8 T + 110 T^{2} + 561 T^{3} + 5136 T^{4} + 26697 T^{5} + 222132 T^{6} + 1395873 T^{7} + 10324443 T^{8} + 60423563 T^{9} + 375353534 T^{10} + 1867603862 T^{11} + 11382213016 T^{12} + 1867603862 p T^{13} + 375353534 p^{2} T^{14} + 60423563 p^{3} T^{15} + 10324443 p^{4} T^{16} + 1395873 p^{5} T^{17} + 222132 p^{6} T^{18} + 26697 p^{7} T^{19} + 5136 p^{8} T^{20} + 561 p^{9} T^{21} + 110 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 8 T + 285 T^{2} + 2328 T^{3} + 41990 T^{4} + 329585 T^{5} + 4122834 T^{6} + 29895592 T^{7} + 295299907 T^{8} + 1931900130 T^{9} + 16051467481 T^{10} + 93483683185 T^{11} + 673998628252 T^{12} + 93483683185 p T^{13} + 16051467481 p^{2} T^{14} + 1931900130 p^{3} T^{15} + 295299907 p^{4} T^{16} + 29895592 p^{5} T^{17} + 4122834 p^{6} T^{18} + 329585 p^{7} T^{19} + 41990 p^{8} T^{20} + 2328 p^{9} T^{21} + 285 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 + T + 230 T^{2} - 257 T^{3} + 24637 T^{4} - 76060 T^{5} + 1796100 T^{6} - 7778661 T^{7} + 108454823 T^{8} - 482085656 T^{9} + 5703268056 T^{10} - 22805439650 T^{11} + 255177914992 T^{12} - 22805439650 p T^{13} + 5703268056 p^{2} T^{14} - 482085656 p^{3} T^{15} + 108454823 p^{4} T^{16} - 7778661 p^{5} T^{17} + 1796100 p^{6} T^{18} - 76060 p^{7} T^{19} + 24637 p^{8} T^{20} - 257 p^{9} T^{21} + 230 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 2 T + 279 T^{2} - 920 T^{3} + 38932 T^{4} - 175056 T^{5} + 3671987 T^{6} - 19664643 T^{7} + 264316714 T^{8} - 1510203737 T^{9} + 15303609823 T^{10} - 85368924579 T^{11} + 725763544528 T^{12} - 85368924579 p T^{13} + 15303609823 p^{2} T^{14} - 1510203737 p^{3} T^{15} + 264316714 p^{4} T^{16} - 19664643 p^{5} T^{17} + 3671987 p^{6} T^{18} - 175056 p^{7} T^{19} + 38932 p^{8} T^{20} - 920 p^{9} T^{21} + 279 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 34 T + 896 T^{2} + 16727 T^{3} + 267326 T^{4} + 3590905 T^{5} + 43102036 T^{6} + 460340779 T^{7} + 4494844159 T^{8} + 40090808867 T^{9} + 331074847308 T^{10} + 2529205380764 T^{11} + 17993329433412 T^{12} + 2529205380764 p T^{13} + 331074847308 p^{2} T^{14} + 40090808867 p^{3} T^{15} + 4494844159 p^{4} T^{16} + 460340779 p^{5} T^{17} + 43102036 p^{6} T^{18} + 3590905 p^{7} T^{19} + 267326 p^{8} T^{20} + 16727 p^{9} T^{21} + 896 p^{10} T^{22} + 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 5 T + 441 T^{2} - 1896 T^{3} + 94214 T^{4} - 351860 T^{5} + 12993530 T^{6} - 42520621 T^{7} + 1294765158 T^{8} - 3740747497 T^{9} + 98490903865 T^{10} - 252621044096 T^{11} + 5872652829352 T^{12} - 252621044096 p T^{13} + 98490903865 p^{2} T^{14} - 3740747497 p^{3} T^{15} + 1294765158 p^{4} T^{16} - 42520621 p^{5} T^{17} + 12993530 p^{6} T^{18} - 351860 p^{7} T^{19} + 94214 p^{8} T^{20} - 1896 p^{9} T^{21} + 441 p^{10} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 26 T + 730 T^{2} + 11536 T^{3} + 185282 T^{4} + 2076094 T^{5} + 23928077 T^{6} + 201664335 T^{7} + 1842506089 T^{8} + 12100923663 T^{9} + 98573053649 T^{10} + 562897464460 T^{11} + 5101887162824 T^{12} + 562897464460 p T^{13} + 98573053649 p^{2} T^{14} + 12100923663 p^{3} T^{15} + 1842506089 p^{4} T^{16} + 201664335 p^{5} T^{17} + 23928077 p^{6} T^{18} + 2076094 p^{7} T^{19} + 185282 p^{8} T^{20} + 11536 p^{9} T^{21} + 730 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 26 T + 752 T^{2} + 13491 T^{3} + 235281 T^{4} + 3266857 T^{5} + 43103756 T^{6} + 492402970 T^{7} + 5338943579 T^{8} + 52076341808 T^{9} + 483789691118 T^{10} + 4119791757429 T^{11} + 33509307831364 T^{12} + 4119791757429 p T^{13} + 483789691118 p^{2} T^{14} + 52076341808 p^{3} T^{15} + 5338943579 p^{4} T^{16} + 492402970 p^{5} T^{17} + 43103756 p^{6} T^{18} + 3266857 p^{7} T^{19} + 235281 p^{8} T^{20} + 13491 p^{9} T^{21} + 752 p^{10} T^{22} + 26 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 6 T + 375 T^{2} + 1475 T^{3} + 71151 T^{4} + 178925 T^{5} + 9349095 T^{6} + 14634395 T^{7} + 969096135 T^{8} + 986114059 T^{9} + 83407729450 T^{10} + 63325311176 T^{11} + 6068485120114 T^{12} + 63325311176 p T^{13} + 83407729450 p^{2} T^{14} + 986114059 p^{3} T^{15} + 969096135 p^{4} T^{16} + 14634395 p^{5} T^{17} + 9349095 p^{6} T^{18} + 178925 p^{7} T^{19} + 71151 p^{8} T^{20} + 1475 p^{9} T^{21} + 375 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 94 T + 4589 T^{2} + 153184 T^{3} + 3904973 T^{4} + 80682013 T^{5} + 1402386991 T^{6} + 21038436416 T^{7} + 277565120451 T^{8} + 3266755261035 T^{9} + 34672343703145 T^{10} + 334475660641729 T^{11} + 2946747864969272 T^{12} + 334475660641729 p T^{13} + 34672343703145 p^{2} T^{14} + 3266755261035 p^{3} T^{15} + 277565120451 p^{4} T^{16} + 21038436416 p^{5} T^{17} + 1402386991 p^{6} T^{18} + 80682013 p^{7} T^{19} + 3904973 p^{8} T^{20} + 153184 p^{9} T^{21} + 4589 p^{10} T^{22} + 94 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 22 T + 668 T^{2} + 9806 T^{3} + 171777 T^{4} + 1817947 T^{5} + 22903099 T^{6} + 169880788 T^{7} + 1651008275 T^{8} + 6379023100 T^{9} + 53269643441 T^{10} - 183530814455 T^{11} + 710939601926 T^{12} - 183530814455 p T^{13} + 53269643441 p^{2} T^{14} + 6379023100 p^{3} T^{15} + 1651008275 p^{4} T^{16} + 169880788 p^{5} T^{17} + 22903099 p^{6} T^{18} + 1817947 p^{7} T^{19} + 171777 p^{8} T^{20} + 9806 p^{9} T^{21} + 668 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 9 T + 367 T^{2} + 2038 T^{3} + 62223 T^{4} + 141043 T^{5} + 6475151 T^{6} - 14731329 T^{7} + 423885699 T^{8} - 4946425456 T^{9} + 15523331202 T^{10} - 8089192135 p T^{11} + 470674369434 T^{12} - 8089192135 p^{2} T^{13} + 15523331202 p^{2} T^{14} - 4946425456 p^{3} T^{15} + 423885699 p^{4} T^{16} - 14731329 p^{5} T^{17} + 6475151 p^{6} T^{18} + 141043 p^{7} T^{19} + 62223 p^{8} T^{20} + 2038 p^{9} T^{21} + 367 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 8 T + 448 T^{2} - 2918 T^{3} + 100176 T^{4} - 424192 T^{5} + 13340059 T^{6} - 13375319 T^{7} + 1118067097 T^{8} + 5037020331 T^{9} + 59946946589 T^{10} + 927626542684 T^{11} + 3267800357708 T^{12} + 927626542684 p T^{13} + 59946946589 p^{2} T^{14} + 5037020331 p^{3} T^{15} + 1118067097 p^{4} T^{16} - 13375319 p^{5} T^{17} + 13340059 p^{6} T^{18} - 424192 p^{7} T^{19} + 100176 p^{8} T^{20} - 2918 p^{9} T^{21} + 448 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 3 T + 589 T^{2} + 1274 T^{3} + 175478 T^{4} + 280583 T^{5} + 35372418 T^{6} + 43704857 T^{7} + 5387584795 T^{8} + 5415512026 T^{9} + 651937772393 T^{10} + 561070696873 T^{11} + 64130514360812 T^{12} + 561070696873 p T^{13} + 651937772393 p^{2} T^{14} + 5415512026 p^{3} T^{15} + 5387584795 p^{4} T^{16} + 43704857 p^{5} T^{17} + 35372418 p^{6} T^{18} + 280583 p^{7} T^{19} + 175478 p^{8} T^{20} + 1274 p^{9} T^{21} + 589 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 29 T + 1249 T^{2} + 25366 T^{3} + 632678 T^{4} + 10020564 T^{5} + 185253972 T^{6} + 2425870975 T^{7} + 36546622034 T^{8} + 409964738669 T^{9} + 5276102462169 T^{10} + 51721589331614 T^{11} + 581746423923696 T^{12} + 51721589331614 p T^{13} + 5276102462169 p^{2} T^{14} + 409964738669 p^{3} T^{15} + 36546622034 p^{4} T^{16} + 2425870975 p^{5} T^{17} + 185253972 p^{6} T^{18} + 10020564 p^{7} T^{19} + 632678 p^{8} T^{20} + 25366 p^{9} T^{21} + 1249 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.01674771650957396072837151365, −2.80294133518885976870544962532, −2.74172504090766112688882154192, −2.69177035856905714814283333913, −2.64756982846664320041589011994, −2.60603707647028374176131832546, −2.53861314066860456593617012536, −2.47953084450563155695449070636, −2.45319269308111346817202783528, −2.37488104136698565565192802108, −2.28853427138631227114008559429, −2.18122377984791653742117822875, −2.03072915276865407080076258786, −1.96143848324252131397511520523, −1.95250967370807271707688542593, −1.82360269012554045815030523647, −1.65848684724560486740068157910, −1.56463360145694659267561460168, −1.55742081105053074740450401679, −1.50826195630414206484880645735, −1.47391494826441337471975387298, −1.38390311239823449143319902628, −1.19247218724192672406703918660, −1.05713099778181009484973797491, −1.02920026592013877964750193533, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1.02920026592013877964750193533, 1.05713099778181009484973797491, 1.19247218724192672406703918660, 1.38390311239823449143319902628, 1.47391494826441337471975387298, 1.50826195630414206484880645735, 1.55742081105053074740450401679, 1.56463360145694659267561460168, 1.65848684724560486740068157910, 1.82360269012554045815030523647, 1.95250967370807271707688542593, 1.96143848324252131397511520523, 2.03072915276865407080076258786, 2.18122377984791653742117822875, 2.28853427138631227114008559429, 2.37488104136698565565192802108, 2.45319269308111346817202783528, 2.47953084450563155695449070636, 2.53861314066860456593617012536, 2.60603707647028374176131832546, 2.64756982846664320041589011994, 2.69177035856905714814283333913, 2.74172504090766112688882154192, 2.80294133518885976870544962532, 3.01674771650957396072837151365
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.