Dirichlet series
| L(s) = 1 | + 2·2-s − 4·3-s − 4-s − 4·5-s − 8·6-s − 6·8-s + 8·9-s − 8·10-s + 4·12-s + 16·15-s − 16-s + 8·17-s + 16·18-s + 4·20-s + 24·24-s + 8·25-s − 16·27-s − 4·29-s + 32·30-s + 8·31-s + 14·32-s + 16·34-s − 8·36-s − 20·37-s + 24·40-s + 16·43-s − 32·45-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s − 1/2·4-s − 1.78·5-s − 3.26·6-s − 2.12·8-s + 8/3·9-s − 2.52·10-s + 1.15·12-s + 4.13·15-s − 1/4·16-s + 1.94·17-s + 3.77·18-s + 0.894·20-s + 4.89·24-s + 8/5·25-s − 3.07·27-s − 0.742·29-s + 5.84·30-s + 1.43·31-s + 2.47·32-s + 2.74·34-s − 4/3·36-s − 3.28·37-s + 3.79·40-s + 2.43·43-s − 4.77·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.62341\times 10^{9}\) |
| Root analytic conductor: | \(2.50205\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.9105249489\) |
| \(L(\frac12)\) | \(\approx\) | \(0.9105249489\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + 5 T^{2} - 3 p T^{3} + 3 p T^{4} - p^{2} T^{5} + p T^{6} - p^{3} T^{7} + 3 p^{3} T^{8} - 3 p^{4} T^{9} + 5 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 34 T^{4} + 16 p T^{5} + 16 p T^{6} + 52 T^{7} + 83 T^{8} + 32 p T^{9} + 8 p^{2} T^{10} + 160 p T^{11} + 1540 T^{12} + 160 p^{2} T^{13} + 8 p^{4} T^{14} + 32 p^{4} T^{15} + 83 p^{4} T^{16} + 52 p^{5} T^{17} + 16 p^{7} T^{18} + 16 p^{8} T^{19} + 34 p^{8} T^{20} + 16 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 4 T + 8 T^{2} + 8 T^{3} - 6 p T^{4} - 96 T^{5} - 112 T^{6} + 52 T^{7} + 723 T^{8} + 552 T^{9} - 1912 T^{10} - 7928 T^{11} - 25276 T^{12} - 7928 p T^{13} - 1912 p^{2} T^{14} + 552 p^{3} T^{15} + 723 p^{4} T^{16} + 52 p^{5} T^{17} - 112 p^{6} T^{18} - 96 p^{7} T^{19} - 6 p^{9} T^{20} + 8 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 32 T^{3} - 206 T^{4} - 160 T^{5} + 512 T^{6} - 3328 T^{7} + 26575 T^{8} + 13056 T^{9} + 11776 T^{10} + 286144 T^{11} - 3400484 T^{12} + 286144 p T^{13} + 11776 p^{2} T^{14} + 13056 p^{3} T^{15} + 26575 p^{4} T^{16} - 3328 p^{5} T^{17} + 512 p^{6} T^{18} - 160 p^{7} T^{19} - 206 p^{8} T^{20} + 32 p^{9} T^{21} + p^{12} T^{24} \) | |
| 13 | \( 1 - 20 T^{3} - 174 T^{4} - 596 T^{5} + 200 T^{6} - 2744 T^{7} + 55763 T^{8} + 14088 T^{9} + 267288 T^{10} - 722392 T^{11} - 4753628 T^{12} - 722392 p T^{13} + 267288 p^{2} T^{14} + 14088 p^{3} T^{15} + 55763 p^{4} T^{16} - 2744 p^{5} T^{17} + 200 p^{6} T^{18} - 596 p^{7} T^{19} - 174 p^{8} T^{20} - 20 p^{9} T^{21} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 4 T + 58 T^{2} - 140 T^{3} + 1383 T^{4} - 1648 T^{5} + 23228 T^{6} - 1648 p T^{7} + 1383 p^{2} T^{8} - 140 p^{3} T^{9} + 58 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 4 p T^{3} - 254 T^{4} - 308 T^{5} + 8 p^{2} T^{6} + 3600 T^{7} + 64723 T^{8} + 135088 T^{9} + 1054584 T^{10} + 4406416 T^{11} - 13090364 T^{12} + 4406416 p T^{13} + 1054584 p^{2} T^{14} + 135088 p^{3} T^{15} + 64723 p^{4} T^{16} + 3600 p^{5} T^{17} + 8 p^{8} T^{18} - 308 p^{7} T^{19} - 254 p^{8} T^{20} + 4 p^{10} T^{21} + p^{12} T^{24} \) | |
| 23 | \( 1 - 140 T^{2} + 10730 T^{4} - 569180 T^{6} + 22853823 T^{8} - 724561688 T^{10} + 18511773292 T^{12} - 724561688 p^{2} T^{14} + 22853823 p^{4} T^{16} - 569180 p^{6} T^{18} + 10730 p^{8} T^{20} - 140 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 + 4 T + 8 T^{2} - 100 T^{3} - 454 T^{4} + 76 p T^{5} + 17448 T^{6} - 18780 T^{7} - 48257 T^{8} + 1942504 T^{9} + 20384272 T^{10} + 8166104 T^{11} - 307305556 T^{12} + 8166104 p T^{13} + 20384272 p^{2} T^{14} + 1942504 p^{3} T^{15} - 48257 p^{4} T^{16} - 18780 p^{5} T^{17} + 17448 p^{6} T^{18} + 76 p^{8} T^{19} - 454 p^{8} T^{20} - 100 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 4 T + 170 T^{2} - 548 T^{3} + 12407 T^{4} - 31840 T^{5} + 502140 T^{6} - 31840 p T^{7} + 12407 p^{2} T^{8} - 548 p^{3} T^{9} + 170 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 20 T + 200 T^{2} + 1676 T^{3} + 13818 T^{4} + 99436 T^{5} + 629608 T^{6} + 3767444 T^{7} + 20529119 T^{8} + 106029960 T^{9} + 572847248 T^{10} + 3054508600 T^{11} + 16972881580 T^{12} + 3054508600 p T^{13} + 572847248 p^{2} T^{14} + 106029960 p^{3} T^{15} + 20529119 p^{4} T^{16} + 3767444 p^{5} T^{17} + 629608 p^{6} T^{18} + 99436 p^{7} T^{19} + 13818 p^{8} T^{20} + 1676 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 372 T^{2} + 66450 T^{4} - 7559588 T^{6} + 611181263 T^{8} - 37032380712 T^{10} + 1725045735036 T^{12} - 37032380712 p^{2} T^{14} + 611181263 p^{4} T^{16} - 7559588 p^{6} T^{18} + 66450 p^{8} T^{20} - 372 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 16 T + 128 T^{2} - 1456 T^{3} + 14962 T^{4} - 80464 T^{5} + 432256 T^{6} - 3242224 T^{7} + 554063 T^{8} + 149895008 T^{9} - 978748672 T^{10} + 10447834784 T^{11} - 101492614052 T^{12} + 10447834784 p T^{13} - 978748672 p^{2} T^{14} + 149895008 p^{3} T^{15} + 554063 p^{4} T^{16} - 3242224 p^{5} T^{17} + 432256 p^{6} T^{18} - 80464 p^{7} T^{19} + 14962 p^{8} T^{20} - 1456 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 8 T + 178 T^{2} + 848 T^{3} + 12375 T^{4} + 40136 T^{5} + 604268 T^{6} + 40136 p T^{7} + 12375 p^{2} T^{8} + 848 p^{3} T^{9} + 178 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 4 T + 8 T^{2} + 36 T^{3} + 186 T^{4} - 11260 T^{5} + 44200 T^{6} - 717636 T^{7} - 3727201 T^{8} + 56691992 T^{9} - 167612656 T^{10} + 731526824 T^{11} + 19642794028 T^{12} + 731526824 p T^{13} - 167612656 p^{2} T^{14} + 56691992 p^{3} T^{15} - 3727201 p^{4} T^{16} - 717636 p^{5} T^{17} + 44200 p^{6} T^{18} - 11260 p^{7} T^{19} + 186 p^{8} T^{20} + 36 p^{9} T^{21} + 8 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 16 T + 128 T^{2} - 1068 T^{3} + 9666 T^{4} - 81916 T^{5} + 643720 T^{6} - 99408 p T^{7} + 56589491 T^{8} - 406593184 T^{9} + 2658851000 T^{10} - 18843939424 T^{11} + 133096304068 T^{12} - 18843939424 p T^{13} + 2658851000 p^{2} T^{14} - 406593184 p^{3} T^{15} + 56589491 p^{4} T^{16} - 99408 p^{6} T^{17} + 643720 p^{6} T^{18} - 81916 p^{7} T^{19} + 9666 p^{8} T^{20} - 1068 p^{9} T^{21} + 128 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 20 T + 200 T^{2} - 1392 T^{3} + 16706 T^{4} - 219096 T^{5} + 2009552 T^{6} - 12282692 T^{7} + 94261875 T^{8} - 1070054824 T^{9} + 10076181640 T^{10} - 58881056712 T^{11} + 342120057220 T^{12} - 58881056712 p T^{13} + 10076181640 p^{2} T^{14} - 1070054824 p^{3} T^{15} + 94261875 p^{4} T^{16} - 12282692 p^{5} T^{17} + 2009552 p^{6} T^{18} - 219096 p^{7} T^{19} + 16706 p^{8} T^{20} - 1392 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 24 T + 288 T^{2} - 40 p T^{3} + 35026 T^{4} - 481096 T^{5} + 5050016 T^{6} - 44472552 T^{7} + 431611055 T^{8} - 4488820656 T^{9} + 41458971456 T^{10} - 331005366640 T^{11} + 2603870224156 T^{12} - 331005366640 p T^{13} + 41458971456 p^{2} T^{14} - 4488820656 p^{3} T^{15} + 431611055 p^{4} T^{16} - 44472552 p^{5} T^{17} + 5050016 p^{6} T^{18} - 481096 p^{7} T^{19} + 35026 p^{8} T^{20} - 40 p^{10} T^{21} + 288 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 420 T^{2} + 86946 T^{4} - 12437204 T^{6} + 1395640463 T^{8} - 128112122952 T^{10} + 9864369161052 T^{12} - 128112122952 p^{2} T^{14} + 1395640463 p^{4} T^{16} - 12437204 p^{6} T^{18} + 86946 p^{8} T^{20} - 420 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 260 T^{2} + 36418 T^{4} - 3914036 T^{6} + 381955535 T^{8} - 34248715528 T^{10} + 2684648528412 T^{12} - 34248715528 p^{2} T^{14} + 381955535 p^{4} T^{16} - 3914036 p^{6} T^{18} + 36418 p^{8} T^{20} - 260 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 - 12 T + 430 T^{2} - 4164 T^{3} + 79599 T^{4} - 615800 T^{5} + 8193220 T^{6} - 615800 p T^{7} + 79599 p^{2} T^{8} - 4164 p^{3} T^{9} + 430 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 20 T + 200 T^{2} - 2776 T^{3} + 14978 T^{4} + 154136 T^{5} - 2225232 T^{6} + 36575116 T^{7} - 391023149 T^{8} + 799427344 T^{9} + 5892039048 T^{10} - 195956951952 T^{11} + 3311983427012 T^{12} - 195956951952 p T^{13} + 5892039048 p^{2} T^{14} + 799427344 p^{3} T^{15} - 391023149 p^{4} T^{16} + 36575116 p^{5} T^{17} - 2225232 p^{6} T^{18} + 154136 p^{7} T^{19} + 14978 p^{8} T^{20} - 2776 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 516 T^{2} + 133186 T^{4} - 22283444 T^{6} + 2738419343 T^{8} - 274144745736 T^{10} + 24925487001372 T^{12} - 274144745736 p^{2} T^{14} + 2738419343 p^{4} T^{16} - 22283444 p^{6} T^{18} + 133186 p^{8} T^{20} - 516 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 + 24 T + 522 T^{2} + 9072 T^{3} + 126023 T^{4} + 1521976 T^{5} + 16411804 T^{6} + 1521976 p T^{7} + 126023 p^{2} T^{8} + 9072 p^{3} T^{9} + 522 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.47249720761624434005972353935, −3.37445086166769063437639314866, −3.08969206648217187763780104682, −3.03092113934425044246645735378, −3.02994039812847438531109751740, −3.02835655042583883039801869893, −2.90980877520529859804689917116, −2.86407167262564373905458675766, −2.55271419548687756985818221407, −2.24198287569163153590351038574, −2.16094457043165057572197527955, −2.14676219387995226788234611697, −2.13548155843500077698967116396, −2.09099771087115350517310177641, −1.73445491856747751941634542474, −1.71095377929267256402306355029, −1.69551579924970737754873145237, −1.25346595960211204817691241437, −1.04697500816027365653043528663, −0.913032474368678230306694346627, −0.869354371355108214558356467390, −0.77101554628477259982192621989, −0.68484437299017415742448186954, −0.39773857421787832328786389966, −0.14137974125465279647032442150, 0.14137974125465279647032442150, 0.39773857421787832328786389966, 0.68484437299017415742448186954, 0.77101554628477259982192621989, 0.869354371355108214558356467390, 0.913032474368678230306694346627, 1.04697500816027365653043528663, 1.25346595960211204817691241437, 1.69551579924970737754873145237, 1.71095377929267256402306355029, 1.73445491856747751941634542474, 2.09099771087115350517310177641, 2.13548155843500077698967116396, 2.14676219387995226788234611697, 2.16094457043165057572197527955, 2.24198287569163153590351038574, 2.55271419548687756985818221407, 2.86407167262564373905458675766, 2.90980877520529859804689917116, 3.02835655042583883039801869893, 3.02994039812847438531109751740, 3.03092113934425044246645735378, 3.08969206648217187763780104682, 3.37445086166769063437639314866, 3.47249720761624434005972353935
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.