Dirichlet series
| L(s) = 1 | + 10·5-s + 36·13-s − 14·17-s − 95·25-s + 108·29-s − 16·37-s − 16·41-s + 235·49-s − 220·53-s + 366·61-s + 360·65-s − 158·73-s − 140·85-s + 396·89-s + 224·97-s − 188·101-s − 296·109-s + 276·113-s + 735·121-s − 1.29e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.08e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2.76·13-s − 0.823·17-s − 3.79·25-s + 3.72·29-s − 0.432·37-s − 0.390·41-s + 4.79·49-s − 4.15·53-s + 6·61-s + 5.53·65-s − 2.16·73-s − 1.64·85-s + 4.44·89-s + 2.30·97-s − 1.86·101-s − 2.71·109-s + 2.44·113-s + 6.07·121-s − 10.3·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 7.44·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 3^{24} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.94722\times 10^{22}\) |
| Root analytic conductor: | \(8.63426\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 3^{24} \cdot 19^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.7007606840\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7007606840\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 19 | \( ( 1 + p T^{2} )^{6} \) | |
| good | 5 | \( ( 1 - p T + 17 p T^{2} - 76 p T^{3} + 4343 T^{4} - 15839 T^{5} + 130694 T^{6} - 15839 p^{2} T^{7} + 4343 p^{4} T^{8} - 76 p^{7} T^{9} + 17 p^{9} T^{10} - p^{11} T^{11} + p^{12} T^{12} )^{2} \) |
| 7 | \( 1 - 235 T^{2} + 4197 p T^{4} - 2663156 T^{6} + 194509433 T^{8} - 11883537777 T^{10} + 623670078502 T^{12} - 11883537777 p^{4} T^{14} + 194509433 p^{8} T^{16} - 2663156 p^{12} T^{18} + 4197 p^{17} T^{20} - 235 p^{20} T^{22} + p^{24} T^{24} \) | |
| 11 | \( 1 - 735 T^{2} + 221 p^{3} T^{4} - 81445924 T^{6} + 17086549253 T^{8} - 2837614900733 T^{10} + 380758281609830 T^{12} - 2837614900733 p^{4} T^{14} + 17086549253 p^{8} T^{16} - 81445924 p^{12} T^{18} + 221 p^{19} T^{20} - 735 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 - 18 T + 70 p T^{2} - 13642 T^{3} + 363023 T^{4} - 334532 p T^{5} + 80366564 T^{6} - 334532 p^{3} T^{7} + 363023 p^{4} T^{8} - 13642 p^{6} T^{9} + 70 p^{9} T^{10} - 18 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 + 7 T + 291 T^{2} + 2612 T^{3} + 152105 T^{4} + 689829 T^{5} + 22565542 T^{6} + 689829 p^{2} T^{7} + 152105 p^{4} T^{8} + 2612 p^{6} T^{9} + 291 p^{8} T^{10} + 7 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 23 | \( 1 + 172 T^{2} + 920546 T^{4} + 155805308 T^{6} + 468751782191 T^{8} + 76691367524952 T^{10} + 158291741754975324 T^{12} + 76691367524952 p^{4} T^{14} + 468751782191 p^{8} T^{16} + 155805308 p^{12} T^{18} + 920546 p^{16} T^{20} + 172 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 - 54 T + 2662 T^{2} - 65374 T^{3} + 35131 p T^{4} + 106468 T^{5} - 510492556 T^{6} + 106468 p^{2} T^{7} + 35131 p^{5} T^{8} - 65374 p^{6} T^{9} + 2662 p^{8} T^{10} - 54 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 956 T^{2} + 435074 T^{4} + 1253267252 T^{6} - 763618616593 T^{8} - 82612284560760 T^{10} + 1560615069505018140 T^{12} - 82612284560760 p^{4} T^{14} - 763618616593 p^{8} T^{16} + 1253267252 p^{12} T^{18} + 435074 p^{16} T^{20} - 956 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 8 T + 4682 T^{2} + 77272 T^{3} + 11920751 T^{4} + 178024608 T^{5} + 20564143980 T^{6} + 178024608 p^{2} T^{7} + 11920751 p^{4} T^{8} + 77272 p^{6} T^{9} + 4682 p^{8} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 8 T + 4266 T^{2} - 92456 T^{3} + 7066751 T^{4} - 477391392 T^{5} + 8526554476 T^{6} - 477391392 p^{2} T^{7} + 7066751 p^{4} T^{8} - 92456 p^{6} T^{9} + 4266 p^{8} T^{10} + 8 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 - 15083 T^{2} + 112846419 T^{4} - 551063467780 T^{6} + 1946818081760585 T^{8} - 5226800226152349249 T^{10} + \)\(10\!\cdots\!22\)\( T^{12} - 5226800226152349249 p^{4} T^{14} + 1946818081760585 p^{8} T^{16} - 551063467780 p^{12} T^{18} + 112846419 p^{16} T^{20} - 15083 p^{20} T^{22} + p^{24} T^{24} \) | |
| 47 | \( 1 - 9363 T^{2} + 50590459 T^{4} - 202383919012 T^{6} + 664789364276321 T^{8} - 1852592670682844297 T^{10} + \)\(44\!\cdots\!58\)\( T^{12} - 1852592670682844297 p^{4} T^{14} + 664789364276321 p^{8} T^{16} - 202383919012 p^{12} T^{18} + 50590459 p^{16} T^{20} - 9363 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 110 T + 12670 T^{2} + 901478 T^{3} + 69276623 T^{4} + 3776995004 T^{5} + 228574826948 T^{6} + 3776995004 p^{2} T^{7} + 69276623 p^{4} T^{8} + 901478 p^{6} T^{9} + 12670 p^{8} T^{10} + 110 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 25084 T^{2} + 310726530 T^{4} - 2535256768844 T^{6} + 15298725524594831 T^{8} - 72526171978175529720 T^{10} + \)\(27\!\cdots\!16\)\( T^{12} - 72526171978175529720 p^{4} T^{14} + 15298725524594831 p^{8} T^{16} - 2535256768844 p^{12} T^{18} + 310726530 p^{16} T^{20} - 25084 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 - 3 p T + 21895 T^{2} - 1691308 T^{3} + 101702309 T^{4} - 5086169837 T^{5} + 275244985862 T^{6} - 5086169837 p^{2} T^{7} + 101702309 p^{4} T^{8} - 1691308 p^{6} T^{9} + 21895 p^{8} T^{10} - 3 p^{11} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 - 20540 T^{2} + 222333762 T^{4} - 1803577565452 T^{6} + 12139539281273039 T^{8} - 1019754636232895784 p T^{10} + \)\(32\!\cdots\!52\)\( T^{12} - 1019754636232895784 p^{5} T^{14} + 12139539281273039 p^{8} T^{16} - 1803577565452 p^{12} T^{18} + 222333762 p^{16} T^{20} - 20540 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( 1 - 30332 T^{2} + 488440610 T^{4} - 5472696693004 T^{6} + 46779283839596207 T^{8} - \)\(31\!\cdots\!96\)\( T^{10} + \)\(17\!\cdots\!12\)\( T^{12} - \)\(31\!\cdots\!96\)\( p^{4} T^{14} + 46779283839596207 p^{8} T^{16} - 5472696693004 p^{12} T^{18} + 488440610 p^{16} T^{20} - 30332 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 + 79 T + 21687 T^{2} + 971780 T^{3} + 193189109 T^{4} + 5616816525 T^{5} + 1160270115814 T^{6} + 5616816525 p^{2} T^{7} + 193189109 p^{4} T^{8} + 971780 p^{6} T^{9} + 21687 p^{8} T^{10} + 79 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 - 39120 T^{2} + 746039530 T^{4} - 9673218910288 T^{6} + 97657155341993039 T^{8} - \)\(80\!\cdots\!96\)\( T^{10} + \)\(55\!\cdots\!44\)\( T^{12} - \)\(80\!\cdots\!96\)\( p^{4} T^{14} + 97657155341993039 p^{8} T^{16} - 9673218910288 p^{12} T^{18} + 746039530 p^{16} T^{20} - 39120 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 - 60992 T^{2} + 1802549738 T^{4} - 34109783125504 T^{6} + 460356300001036079 T^{8} - \)\(46\!\cdots\!08\)\( T^{10} + \)\(36\!\cdots\!60\)\( T^{12} - \)\(46\!\cdots\!08\)\( p^{4} T^{14} + 460356300001036079 p^{8} T^{16} - 34109783125504 p^{12} T^{18} + 1802549738 p^{16} T^{20} - 60992 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( ( 1 - 198 T + 47574 T^{2} - 6283566 T^{3} + 920160639 T^{4} - 90651938220 T^{5} + 9589245348116 T^{6} - 90651938220 p^{2} T^{7} + 920160639 p^{4} T^{8} - 6283566 p^{6} T^{9} + 47574 p^{8} T^{10} - 198 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 112 T + 38098 T^{2} - 3180016 T^{3} + 637160879 T^{4} - 41524383136 T^{5} + 6898094098940 T^{6} - 41524383136 p^{2} T^{7} + 637160879 p^{4} T^{8} - 3180016 p^{6} T^{9} + 38098 p^{8} T^{10} - 112 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−2.32716935548829921573177326141, −2.32207814585873692514153419569, −2.32180423382941670360975069803, −2.25592359242648779726586414598, −2.12535139297494721356185204158, −2.07842583252219965434712593637, −1.98033706755733009367919555902, −1.95390543034345822906236576246, −1.86961748556256330586314353711, −1.60279381982871404346702778214, −1.55623778703677345027567789669, −1.49757327837425730941318960662, −1.46743088355247677792903385028, −1.24460322356283425327750806793, −1.12292604232282055657852989803, −1.11883178722908233281967671872, −1.00410622157216999863226767527, −0.980413127314194306548945630384, −0.854620014075812625217784240138, −0.68753873819545242601733972737, −0.66631151914075605856333533106, −0.34483588128220280039555110180, −0.30180527423655070099597963398, −0.24108830585436803328977178994, −0.02129274026615008371146289836, 0.02129274026615008371146289836, 0.24108830585436803328977178994, 0.30180527423655070099597963398, 0.34483588128220280039555110180, 0.66631151914075605856333533106, 0.68753873819545242601733972737, 0.854620014075812625217784240138, 0.980413127314194306548945630384, 1.00410622157216999863226767527, 1.11883178722908233281967671872, 1.12292604232282055657852989803, 1.24460322356283425327750806793, 1.46743088355247677792903385028, 1.49757327837425730941318960662, 1.55623778703677345027567789669, 1.60279381982871404346702778214, 1.86961748556256330586314353711, 1.95390543034345822906236576246, 1.98033706755733009367919555902, 2.07842583252219965434712593637, 2.12535139297494721356185204158, 2.25592359242648779726586414598, 2.32180423382941670360975069803, 2.32207814585873692514153419569, 2.32716935548829921573177326141
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.