Dirichlet series
| L(s) = 1 | − 2·3-s + 6·4-s + 2·9-s − 4·11-s − 12·12-s + 12·13-s + 13·16-s + 6·17-s + 12·23-s + 4·27-s − 6·29-s − 6·31-s + 8·33-s + 12·36-s + 18·37-s − 24·39-s + 6·41-s − 24·44-s + 36·47-s − 26·48-s − 12·51-s + 72·52-s − 8·61-s + 20·67-s + 36·68-s − 24·69-s − 20·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3·4-s + 2/3·9-s − 1.20·11-s − 3.46·12-s + 3.32·13-s + 13/4·16-s + 1.45·17-s + 2.50·23-s + 0.769·27-s − 1.11·29-s − 1.07·31-s + 1.39·33-s + 2·36-s + 2.95·37-s − 3.84·39-s + 0.937·41-s − 3.61·44-s + 5.25·47-s − 3.75·48-s − 1.68·51-s + 9.98·52-s − 1.02·61-s + 2.44·67-s + 4.36·68-s − 2.88·69-s − 2.37·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 17^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 17^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.33341\times 10^{6}\) |
| Root analytic conductor: | \(1.84218\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{24} \cdot 17^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.282405826\) |
| \(L(\frac12)\) | \(\approx\) | \(2.282405826\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - 6 T - 2 T^{2} + 138 T^{3} - 353 T^{4} - 1508 T^{5} + 13604 T^{6} - 1508 p T^{7} - 353 p^{2} T^{8} + 138 p^{3} T^{9} - 2 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - 3 p T^{2} + 23 T^{4} - 15 p^{2} T^{6} + 119 T^{8} - 105 p T^{10} + 369 T^{12} - 105 p^{3} T^{14} + 119 p^{4} T^{16} - 15 p^{8} T^{18} + 23 p^{8} T^{20} - 3 p^{11} T^{22} + p^{12} T^{24} \) |
| 3 | \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 11 T^{4} - 10 T^{5} + 10 T^{6} - 10 T^{7} + 10 T^{8} + 40 T^{9} + 260 T^{10} + 110 T^{11} + 145 T^{12} + 110 p T^{13} + 260 p^{2} T^{14} + 40 p^{3} T^{15} + 10 p^{4} T^{16} - 10 p^{5} T^{17} + 10 p^{6} T^{18} - 10 p^{7} T^{19} - 11 p^{8} T^{20} - 4 p^{9} T^{21} + 2 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 20 T^{3} + 9 T^{4} + 74 T^{5} + 200 T^{6} + 1340 T^{7} + 2890 T^{8} + 2346 T^{9} + 27738 T^{10} + 55770 T^{11} + 32705 T^{12} + 55770 p T^{13} + 27738 p^{2} T^{14} + 2346 p^{3} T^{15} + 2890 p^{4} T^{16} + 1340 p^{5} T^{17} + 200 p^{6} T^{18} + 74 p^{7} T^{19} + 9 p^{8} T^{20} + 20 p^{9} T^{21} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T + 8 T^{2} - 32 T^{3} - 10 T^{4} + 312 T^{5} + 1840 T^{6} - 5924 T^{7} - 14649 T^{8} + 21312 T^{9} + 459080 T^{10} + 227632 T^{11} + 365512 T^{12} + 227632 p T^{13} + 459080 p^{2} T^{14} + 21312 p^{3} T^{15} - 14649 p^{4} T^{16} - 5924 p^{5} T^{17} + 1840 p^{6} T^{18} + 312 p^{7} T^{19} - 10 p^{8} T^{20} - 32 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( ( 1 - 6 T + 59 T^{2} - 246 T^{3} + 1518 T^{4} - 5204 T^{5} + 24675 T^{6} - 5204 p T^{7} + 1518 p^{2} T^{8} - 246 p^{3} T^{9} + 59 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 8 p T^{2} + 11346 T^{4} - 554080 T^{6} + 19718655 T^{8} - 537479512 T^{10} + 11489620204 T^{12} - 537479512 p^{2} T^{14} + 19718655 p^{4} T^{16} - 554080 p^{6} T^{18} + 11346 p^{8} T^{20} - 8 p^{11} T^{22} + p^{12} T^{24} \) | |
| 23 | \( 1 - 12 T + 72 T^{2} - 416 T^{3} + 2554 T^{4} - 9840 T^{5} + 20720 T^{6} + 17060 T^{7} - 533265 T^{8} + 4578680 T^{9} - 28152120 T^{10} + 143963800 T^{11} - 648463920 T^{12} + 143963800 p T^{13} - 28152120 p^{2} T^{14} + 4578680 p^{3} T^{15} - 533265 p^{4} T^{16} + 17060 p^{5} T^{17} + 20720 p^{6} T^{18} - 9840 p^{7} T^{19} + 2554 p^{8} T^{20} - 416 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + 3126 T^{4} + 13374 T^{5} + 39114 T^{6} + 354462 T^{7} + 4092431 T^{8} + 15189444 T^{9} + 46310868 T^{10} + 425042508 T^{11} + 3894084004 T^{12} + 425042508 p T^{13} + 46310868 p^{2} T^{14} + 15189444 p^{3} T^{15} + 4092431 p^{4} T^{16} + 354462 p^{5} T^{17} + 39114 p^{6} T^{18} + 13374 p^{7} T^{19} + 3126 p^{8} T^{20} + 6 p^{10} T^{21} + 18 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 6 T + 18 T^{2} + 48 T^{3} + 1665 T^{4} + 11534 T^{5} + 40386 T^{6} + 66414 T^{7} + 419086 T^{8} + 9419944 T^{9} + 51437876 T^{10} + 147391674 T^{11} - 369798559 T^{12} + 147391674 p T^{13} + 51437876 p^{2} T^{14} + 9419944 p^{3} T^{15} + 419086 p^{4} T^{16} + 66414 p^{5} T^{17} + 40386 p^{6} T^{18} + 11534 p^{7} T^{19} + 1665 p^{8} T^{20} + 48 p^{9} T^{21} + 18 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 18 T + 162 T^{2} - 1150 T^{3} + 3942 T^{4} + 24386 T^{5} - 416302 T^{6} + 3817094 T^{7} - 22764977 T^{8} + 53351700 T^{9} + 276338556 T^{10} - 5112486108 T^{11} + 42751068548 T^{12} - 5112486108 p T^{13} + 276338556 p^{2} T^{14} + 53351700 p^{3} T^{15} - 22764977 p^{4} T^{16} + 3817094 p^{5} T^{17} - 416302 p^{6} T^{18} + 24386 p^{7} T^{19} + 3942 p^{8} T^{20} - 1150 p^{9} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 6 T + 18 T^{2} - 2 T^{3} + 790 T^{4} - 18458 T^{5} + 96530 T^{6} - 861974 T^{7} + 3652191 T^{8} - 10785268 T^{9} + 94786300 T^{10} - 1283624948 T^{11} + 16786117092 T^{12} - 1283624948 p T^{13} + 94786300 p^{2} T^{14} - 10785268 p^{3} T^{15} + 3652191 p^{4} T^{16} - 861974 p^{5} T^{17} + 96530 p^{6} T^{18} - 18458 p^{7} T^{19} + 790 p^{8} T^{20} - 2 p^{9} T^{21} + 18 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 244 T^{2} + 28258 T^{4} - 2098860 T^{6} + 114290639 T^{8} - 5110097840 T^{10} + 216451349804 T^{12} - 5110097840 p^{2} T^{14} + 114290639 p^{4} T^{16} - 2098860 p^{6} T^{18} + 28258 p^{8} T^{20} - 244 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( ( 1 - 18 T + 212 T^{2} - 1550 T^{3} + 11555 T^{4} - 76988 T^{5} + 591304 T^{6} - 76988 p T^{7} + 11555 p^{2} T^{8} - 1550 p^{3} T^{9} + 212 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 - 294 T^{2} + 45333 T^{4} - 4906770 T^{6} + 415676874 T^{8} - 28839408590 T^{10} + 1669465258669 T^{12} - 28839408590 p^{2} T^{14} + 415676874 p^{4} T^{16} - 4906770 p^{6} T^{18} + 45333 p^{8} T^{20} - 294 p^{10} T^{22} + p^{12} T^{24} \) | |
| 59 | \( 1 - 252 T^{2} + 30346 T^{4} - 2538820 T^{6} + 189240895 T^{8} - 13886009792 T^{10} + 904669570204 T^{12} - 13886009792 p^{2} T^{14} + 189240895 p^{4} T^{16} - 2538820 p^{6} T^{18} + 30346 p^{8} T^{20} - 252 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( 1 + 8 T + 32 T^{2} - 552 T^{3} - 8074 T^{4} - 65832 T^{5} - 115936 T^{6} + 3041736 T^{7} + 36019519 T^{8} + 200321488 T^{9} + 1767872 T^{10} - 8574393744 T^{11} - 119658423436 T^{12} - 8574393744 p T^{13} + 1767872 p^{2} T^{14} + 200321488 p^{3} T^{15} + 36019519 p^{4} T^{16} + 3041736 p^{5} T^{17} - 115936 p^{6} T^{18} - 65832 p^{7} T^{19} - 8074 p^{8} T^{20} - 552 p^{9} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 - 10 T + 338 T^{2} - 2902 T^{3} + 51527 T^{4} - 362788 T^{5} + 4461756 T^{6} - 362788 p T^{7} + 51527 p^{2} T^{8} - 2902 p^{3} T^{9} + 338 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 20 T + 200 T^{2} + 2024 T^{3} + 25405 T^{4} + 279682 T^{5} + 2560928 T^{6} + 26611232 T^{7} + 302022294 T^{8} + 2705613542 T^{9} + 21619580330 T^{10} + 199200807274 T^{11} + 1825753645641 T^{12} + 199200807274 p T^{13} + 21619580330 p^{2} T^{14} + 2705613542 p^{3} T^{15} + 302022294 p^{4} T^{16} + 26611232 p^{5} T^{17} + 2560928 p^{6} T^{18} + 279682 p^{7} T^{19} + 25405 p^{8} T^{20} + 2024 p^{9} T^{21} + 200 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 18 T + 162 T^{2} + 70 T^{3} - 14730 T^{4} - 172282 T^{5} - 712366 T^{6} + 6164282 T^{7} + 152877023 T^{8} + 1345139588 T^{9} + 4225327180 T^{10} - 62449346140 T^{11} - 929304648668 T^{12} - 62449346140 p T^{13} + 4225327180 p^{2} T^{14} + 1345139588 p^{3} T^{15} + 152877023 p^{4} T^{16} + 6164282 p^{5} T^{17} - 712366 p^{6} T^{18} - 172282 p^{7} T^{19} - 14730 p^{8} T^{20} + 70 p^{9} T^{21} + 162 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 14 T + 98 T^{2} + 1696 T^{3} + 15181 T^{4} + 21226 T^{5} + 247634 T^{6} + 3341998 T^{7} - 49244934 T^{8} - 649664704 T^{9} - 2135333932 T^{10} - 43163107378 T^{11} - 726637664151 T^{12} - 43163107378 p T^{13} - 2135333932 p^{2} T^{14} - 649664704 p^{3} T^{15} - 49244934 p^{4} T^{16} + 3341998 p^{5} T^{17} + 247634 p^{6} T^{18} + 21226 p^{7} T^{19} + 15181 p^{8} T^{20} + 1696 p^{9} T^{21} + 98 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 620 T^{2} + 185134 T^{4} - 36084060 T^{6} + 5218206815 T^{8} - 595109341720 T^{10} + 54814722227780 T^{12} - 595109341720 p^{2} T^{14} + 5218206815 p^{4} T^{16} - 36084060 p^{6} T^{18} + 185134 p^{8} T^{20} - 620 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( ( 1 + 12 T + 450 T^{2} + 4580 T^{3} + 89651 T^{4} + 748480 T^{5} + 10238416 T^{6} + 748480 p T^{7} + 89651 p^{2} T^{8} + 4580 p^{3} T^{9} + 450 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 52 T + 1352 T^{2} + 25420 T^{3} + 392202 T^{4} + 5121036 T^{5} + 59124968 T^{6} + 637988404 T^{7} + 6714115663 T^{8} + 70975600280 T^{9} + 754486224976 T^{10} + 7917294022312 T^{11} + 80025007432268 T^{12} + 7917294022312 p T^{13} + 754486224976 p^{2} T^{14} + 70975600280 p^{3} T^{15} + 6714115663 p^{4} T^{16} + 637988404 p^{5} T^{17} + 59124968 p^{6} T^{18} + 5121036 p^{7} T^{19} + 392202 p^{8} T^{20} + 25420 p^{9} T^{21} + 1352 p^{10} T^{22} + 52 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.77383242639562350000789181478, −3.72802802400603340031608880037, −3.48875523758687329252596456282, −3.43778967212871711027567830674, −3.32409029753353059268995043521, −3.02745692945773412426452540796, −2.86864461880278934357993649596, −2.81999147830118938369297595604, −2.81962901234300191851026446592, −2.72580324467043436304875022271, −2.69414271355665950304829884478, −2.56506965505868199283080028175, −2.53470155161430605108931351630, −2.40725948490643318158253827361, −2.18540670029300567768617695256, −1.86495210567774231037614642957, −1.81281971701755226449434383201, −1.62662430275646492307907415136, −1.41288717908302102739439640599, −1.33355272478612962401680801780, −1.26845639353507547550243170261, −1.07038090426878260619949525002, −1.00925039481378464120429857240, −0.825044875992226482575245138535, −0.13542850571710197127068292406, 0.13542850571710197127068292406, 0.825044875992226482575245138535, 1.00925039481378464120429857240, 1.07038090426878260619949525002, 1.26845639353507547550243170261, 1.33355272478612962401680801780, 1.41288717908302102739439640599, 1.62662430275646492307907415136, 1.81281971701755226449434383201, 1.86495210567774231037614642957, 2.18540670029300567768617695256, 2.40725948490643318158253827361, 2.53470155161430605108931351630, 2.56506965505868199283080028175, 2.69414271355665950304829884478, 2.72580324467043436304875022271, 2.81962901234300191851026446592, 2.81999147830118938369297595604, 2.86864461880278934357993649596, 3.02745692945773412426452540796, 3.32409029753353059268995043521, 3.43778967212871711027567830674, 3.48875523758687329252596456282, 3.72802802400603340031608880037, 3.77383242639562350000789181478
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.