Dirichlet series
| L(s) = 1 | + 24·2-s + 192·4-s − 718·5-s − 1.47e3·7-s − 1.02e3·8-s − 1.72e4·10-s + 208·11-s + 1.93e4·13-s − 3.53e4·14-s − 3.68e4·16-s − 1.92e4·17-s − 2.54e4·19-s − 1.37e5·20-s + 4.99e3·22-s − 6.74e4·23-s + 3.27e5·25-s + 4.65e5·26-s − 2.82e5·28-s + 3.34e4·29-s + 7.84e4·31-s − 2.94e5·32-s − 4.61e5·34-s + 1.05e6·35-s − 4.96e5·37-s − 6.10e5·38-s + 7.35e5·40-s + 1.95e6·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 2.56·5-s − 1.62·7-s − 0.707·8-s − 5.44·10-s + 0.0471·11-s + 2.44·13-s − 3.43·14-s − 9/4·16-s − 0.950·17-s − 0.850·19-s − 3.85·20-s + 0.0999·22-s − 1.15·23-s + 4.19·25-s + 5.19·26-s − 2.43·28-s + 0.254·29-s + 0.472·31-s − 1.59·32-s − 2.01·34-s + 4.16·35-s − 1.61·37-s − 1.80·38-s + 1.81·40-s + 4.42·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{12} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.71847\times 10^{9}\) |
| Root analytic conductor: | \(6.27379\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.8219756335\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8219756335\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T + p^{6} T^{2} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 + 1471 T + 86012 p T^{2} + 80485 p^{3} T^{3} + 86012 p^{8} T^{4} + 1471 p^{14} T^{5} + p^{21} T^{6} \) | |
| good | 5 | \( 1 + 718 T + 37508 p T^{2} + 13671352 T^{3} + 2315957816 T^{4} + 792702977894 p T^{5} + 68165690531206 p^{2} T^{6} + 792702977894 p^{8} T^{7} + 2315957816 p^{14} T^{8} + 13671352 p^{21} T^{9} + 37508 p^{29} T^{10} + 718 p^{35} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 - 208 T - 16052456 T^{2} - 44055040576 T^{3} - 50354304084856 T^{4} + 34463627672265232 p T^{5} + 83882262054616957222 p^{2} T^{6} + 34463627672265232 p^{8} T^{7} - 50354304084856 p^{14} T^{8} - 44055040576 p^{21} T^{9} - 16052456 p^{28} T^{10} - 208 p^{35} T^{11} + p^{42} T^{12} \) | |
| 13 | \( ( 1 - 9697 T + 178266151 T^{2} - 1015849655102 T^{3} + 178266151 p^{7} T^{4} - 9697 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 17 | \( 1 + 1132 p T - 18255559 p T^{2} - 19109086571836 T^{3} - 7405492060721510 p T^{4} + \)\(20\!\cdots\!12\)\( p T^{5} + \)\(14\!\cdots\!33\)\( T^{6} + \)\(20\!\cdots\!12\)\( p^{8} T^{7} - 7405492060721510 p^{15} T^{8} - 19109086571836 p^{21} T^{9} - 18255559 p^{29} T^{10} + 1132 p^{36} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 25419 T - 1819555500 T^{2} - 15387477866581 T^{3} + 2883443547420489312 T^{4} + \)\(45\!\cdots\!67\)\( T^{5} - \)\(29\!\cdots\!62\)\( T^{6} + \)\(45\!\cdots\!67\)\( p^{7} T^{7} + 2883443547420489312 p^{14} T^{8} - 15387477866581 p^{21} T^{9} - 1819555500 p^{28} T^{10} + 25419 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 + 67400 T - 2013922001 T^{2} - 37286894218456 T^{3} + 4516436438752337354 T^{4} - \)\(65\!\cdots\!72\)\( T^{5} - \)\(62\!\cdots\!17\)\( T^{6} - \)\(65\!\cdots\!72\)\( p^{7} T^{7} + 4516436438752337354 p^{14} T^{8} - 37286894218456 p^{21} T^{9} - 2013922001 p^{28} T^{10} + 67400 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( ( 1 - 16738 T + 1607521728 p T^{2} - 462047797169092 T^{3} + 1607521728 p^{8} T^{4} - 16738 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 31 | \( 1 - 78449 T - 21282703899 T^{2} + 14041946279888478 T^{3} - \)\(63\!\cdots\!61\)\( T^{4} - \)\(16\!\cdots\!29\)\( T^{5} + \)\(81\!\cdots\!62\)\( T^{6} - \)\(16\!\cdots\!29\)\( p^{7} T^{7} - \)\(63\!\cdots\!61\)\( p^{14} T^{8} + 14041946279888478 p^{21} T^{9} - 21282703899 p^{28} T^{10} - 78449 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 496599 T + 55080777450 T^{2} - 63443092202259619 T^{3} - \)\(20\!\cdots\!48\)\( T^{4} + \)\(16\!\cdots\!11\)\( T^{5} + \)\(22\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!11\)\( p^{7} T^{7} - \)\(20\!\cdots\!48\)\( p^{14} T^{8} - 63443092202259619 p^{21} T^{9} + 55080777450 p^{28} T^{10} + 496599 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( ( 1 - 976584 T + 666376203147 T^{2} - 337071452221974288 T^{3} + 666376203147 p^{7} T^{4} - 976584 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 43 | \( ( 1 - 348735 T + 700054671801 T^{2} - 151535711521133642 T^{3} + 700054671801 p^{7} T^{4} - 348735 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 47 | \( 1 - 701328 T - 115309767741 T^{2} + 674905136253738864 T^{3} - \)\(32\!\cdots\!46\)\( T^{4} - \)\(82\!\cdots\!32\)\( T^{5} + \)\(24\!\cdots\!47\)\( T^{6} - \)\(82\!\cdots\!32\)\( p^{7} T^{7} - \)\(32\!\cdots\!46\)\( p^{14} T^{8} + 674905136253738864 p^{21} T^{9} - 115309767741 p^{28} T^{10} - 701328 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 2917374 T + 2276248194948 T^{2} - 2915903692248000504 T^{3} + \)\(92\!\cdots\!32\)\( T^{4} - \)\(89\!\cdots\!30\)\( T^{5} + \)\(41\!\cdots\!10\)\( T^{6} - \)\(89\!\cdots\!30\)\( p^{7} T^{7} + \)\(92\!\cdots\!32\)\( p^{14} T^{8} - 2915903692248000504 p^{21} T^{9} + 2276248194948 p^{28} T^{10} - 2917374 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 + 492040 T - 1935722678552 T^{2} - 4205359362754399664 T^{3} - \)\(19\!\cdots\!64\)\( T^{4} + \)\(34\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!26\)\( T^{6} + \)\(34\!\cdots\!44\)\( p^{7} T^{7} - \)\(19\!\cdots\!64\)\( p^{14} T^{8} - 4205359362754399664 p^{21} T^{9} - 1935722678552 p^{28} T^{10} + 492040 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 3649370 T + 21457547001 p T^{2} + 1172801966309751426 T^{3} + \)\(24\!\cdots\!42\)\( T^{4} - \)\(30\!\cdots\!02\)\( T^{5} - \)\(16\!\cdots\!35\)\( T^{6} - \)\(30\!\cdots\!02\)\( p^{7} T^{7} + \)\(24\!\cdots\!42\)\( p^{14} T^{8} + 1172801966309751426 p^{21} T^{9} + 21457547001 p^{29} T^{10} - 3649370 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 3647153 T - 6765687171700 T^{2} - 14003623555671703551 T^{3} + \)\(12\!\cdots\!32\)\( T^{4} + \)\(12\!\cdots\!21\)\( T^{5} - \)\(58\!\cdots\!94\)\( T^{6} + \)\(12\!\cdots\!21\)\( p^{7} T^{7} + \)\(12\!\cdots\!32\)\( p^{14} T^{8} - 14003623555671703551 p^{21} T^{9} - 6765687171700 p^{28} T^{10} + 3647153 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( ( 1 - 1770608 T + 10215747485541 T^{2} - 21266983506803318816 T^{3} + 10215747485541 p^{7} T^{4} - 1770608 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 73 | \( 1 + 6183557 T + 1555714548086 T^{2} - 5574639015136889997 T^{3} + \)\(22\!\cdots\!00\)\( T^{4} + \)\(53\!\cdots\!81\)\( T^{5} - \)\(33\!\cdots\!52\)\( T^{6} + \)\(53\!\cdots\!81\)\( p^{7} T^{7} + \)\(22\!\cdots\!00\)\( p^{14} T^{8} - 5574639015136889997 p^{21} T^{9} + 1555714548086 p^{28} T^{10} + 6183557 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 12453589 T + 77848697777253 T^{2} + \)\(26\!\cdots\!18\)\( T^{3} + \)\(16\!\cdots\!99\)\( T^{4} - \)\(55\!\cdots\!71\)\( T^{5} - \)\(37\!\cdots\!70\)\( T^{6} - \)\(55\!\cdots\!71\)\( p^{7} T^{7} + \)\(16\!\cdots\!99\)\( p^{14} T^{8} + \)\(26\!\cdots\!18\)\( p^{21} T^{9} + 77848697777253 p^{28} T^{10} + 12453589 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( ( 1 - 5805560 T + 38113159247016 T^{2} - 87016548674576961368 T^{3} + 38113159247016 p^{7} T^{4} - 5805560 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
| 89 | \( 1 + 8292612 T - 649109442279 p T^{2} - \)\(18\!\cdots\!76\)\( T^{3} + \)\(53\!\cdots\!94\)\( T^{4} + \)\(13\!\cdots\!92\)\( T^{5} - \)\(28\!\cdots\!07\)\( T^{6} + \)\(13\!\cdots\!92\)\( p^{7} T^{7} + \)\(53\!\cdots\!94\)\( p^{14} T^{8} - \)\(18\!\cdots\!76\)\( p^{21} T^{9} - 649109442279 p^{29} T^{10} + 8292612 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( ( 1 + 27974764 T + 374108696829248 T^{2} + \)\(35\!\cdots\!90\)\( T^{3} + 374108696829248 p^{7} T^{4} + 27974764 p^{14} T^{5} + p^{21} T^{6} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.83634801416728199777581142563, −5.78777072906204979094153741200, −5.54522711731269340978223923840, −5.52964110576726671359675751837, −5.15923355520160377862754706903, −4.83011641578188884756396293243, −4.42686203323321225932782687101, −4.23732628173985976310551860658, −4.15465608214738534476864113060, −4.04357381744090621800509854284, −3.92258283357970668948444383882, −3.78714699434622072932626696166, −3.76698029037631391237237910718, −3.16122533120346607465314521802, −2.91090923151674030626595992115, −2.85191708908294916138026113073, −2.55086593283578715974973612680, −2.50663626104104335141945779327, −2.01522434079631266415001625044, −1.40901703197005777638183961800, −1.17863234279105396416378251299, −0.889149048579573550153199036303, −0.71581401436188315410771316539, −0.32789551687188016673700767168, −0.098862786074179406825518645034, 0.098862786074179406825518645034, 0.32789551687188016673700767168, 0.71581401436188315410771316539, 0.889149048579573550153199036303, 1.17863234279105396416378251299, 1.40901703197005777638183961800, 2.01522434079631266415001625044, 2.50663626104104335141945779327, 2.55086593283578715974973612680, 2.85191708908294916138026113073, 2.91090923151674030626595992115, 3.16122533120346607465314521802, 3.76698029037631391237237910718, 3.78714699434622072932626696166, 3.92258283357970668948444383882, 4.04357381744090621800509854284, 4.15465608214738534476864113060, 4.23732628173985976310551860658, 4.42686203323321225932782687101, 4.83011641578188884756396293243, 5.15923355520160377862754706903, 5.52964110576726671359675751837, 5.54522711731269340978223923840, 5.78777072906204979094153741200, 5.83634801416728199777581142563
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.