Dirichlet series
| L(s) = 1 | − 2·2-s + 3·4-s − 8·5-s + 16·10-s − 24·13-s − 7·16-s + 40·17-s − 24·20-s − 28·25-s + 48·26-s − 72·29-s − 14·32-s − 80·34-s − 88·37-s + 200·41-s − 42·49-s + 56·50-s − 72·52-s − 104·53-s + 144·58-s + 104·61-s + 31·64-s + 192·65-s + 120·68-s + 312·73-s + 176·74-s + 56·80-s + ⋯ |
| L(s) = 1 | − 2-s + 3/4·4-s − 8/5·5-s + 8/5·10-s − 1.84·13-s − 0.437·16-s + 2.35·17-s − 6/5·20-s − 1.11·25-s + 1.84·26-s − 2.48·29-s − 0.437·32-s − 2.35·34-s − 2.37·37-s + 4.87·41-s − 6/7·49-s + 1.11·50-s − 1.38·52-s − 1.96·53-s + 2.48·58-s + 1.70·61-s + 0.484·64-s + 2.95·65-s + 1.76·68-s + 4.27·73-s + 2.37·74-s + 7/10·80-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09856\times 10^{10}\) |
| Root analytic conductor: | \(2.62040\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.331332716\) |
| \(L(\frac12)\) | \(\approx\) | \(1.331332716\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + p T + T^{2} - p^{2} T^{3} - p^{2} T^{4} + p^{5} T^{5} + 5 p^{4} T^{6} + p^{7} T^{7} - p^{6} T^{8} - p^{8} T^{9} + p^{8} T^{10} + p^{11} T^{11} + p^{12} T^{12} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + p T^{2} )^{6} \) | |
| good | 5 | \( ( 1 + 4 T + 38 T^{2} + 116 T^{3} + 1151 T^{4} + 4008 T^{5} + 44052 T^{6} + 4008 p^{2} T^{7} + 1151 p^{4} T^{8} + 116 p^{6} T^{9} + 38 p^{8} T^{10} + 4 p^{10} T^{11} + p^{12} T^{12} )^{2} \) |
| 11 | \( 1 - 740 T^{2} + 281730 T^{4} - 73864084 T^{6} + 14875387919 T^{8} - 2411663232072 T^{10} + 321087036485788 T^{12} - 2411663232072 p^{4} T^{14} + 14875387919 p^{8} T^{16} - 73864084 p^{12} T^{18} + 281730 p^{16} T^{20} - 740 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 12 T + 610 T^{2} + 6844 T^{3} + 198431 T^{4} + 1981208 T^{5} + 40776860 T^{6} + 1981208 p^{2} T^{7} + 198431 p^{4} T^{8} + 6844 p^{6} T^{9} + 610 p^{8} T^{10} + 12 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 17 | \( ( 1 - 20 T + 438 T^{2} - 1924 T^{3} + 104879 T^{4} - 2682408 T^{5} + 75178804 T^{6} - 2682408 p^{2} T^{7} + 104879 p^{4} T^{8} - 1924 p^{6} T^{9} + 438 p^{8} T^{10} - 20 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 - 2764 T^{2} + 3455010 T^{4} - 2609922140 T^{6} + 1366901328335 T^{8} - 560511269087640 T^{10} + 206098561542183772 T^{12} - 560511269087640 p^{4} T^{14} + 1366901328335 p^{8} T^{16} - 2609922140 p^{12} T^{18} + 3455010 p^{16} T^{20} - 2764 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 - 2020 T^{2} + 2623266 T^{4} - 2634129236 T^{6} + 2098673243183 T^{8} - 1406916031708488 T^{10} + 807693594996846556 T^{12} - 1406916031708488 p^{4} T^{14} + 2098673243183 p^{8} T^{16} - 2634129236 p^{12} T^{18} + 2623266 p^{16} T^{20} - 2020 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 + 36 T + 3234 T^{2} + 127540 T^{3} + 5501919 T^{4} + 190467528 T^{5} + 5856614172 T^{6} + 190467528 p^{2} T^{7} + 5501919 p^{4} T^{8} + 127540 p^{6} T^{9} + 3234 p^{8} T^{10} + 36 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 31 | \( 1 - 6092 T^{2} + 19019970 T^{4} - 40472674204 T^{6} + 65359520265455 T^{8} - 84287994066953880 T^{10} + 89086270372746101404 T^{12} - 84287994066953880 p^{4} T^{14} + 65359520265455 p^{8} T^{16} - 40472674204 p^{12} T^{18} + 19019970 p^{16} T^{20} - 6092 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 44 T + 5330 T^{2} + 244636 T^{3} + 15541631 T^{4} + 571448664 T^{5} + 27550657212 T^{6} + 571448664 p^{2} T^{7} + 15541631 p^{4} T^{8} + 244636 p^{6} T^{9} + 5330 p^{8} T^{10} + 44 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 100 T + 11126 T^{2} - 775316 T^{3} + 48514127 T^{4} - 2496660360 T^{5} + 109734244404 T^{6} - 2496660360 p^{2} T^{7} + 48514127 p^{4} T^{8} - 775316 p^{6} T^{9} + 11126 p^{8} T^{10} - 100 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 43 | \( 1 - 8620 T^{2} + 26632098 T^{4} - 30264260924 T^{6} + 21737997809039 T^{8} - 181921081055514456 T^{10} + \)\(57\!\cdots\!88\)\( T^{12} - 181921081055514456 p^{4} T^{14} + 21737997809039 p^{8} T^{16} - 30264260924 p^{12} T^{18} + 26632098 p^{16} T^{20} - 8620 p^{20} T^{22} + p^{24} T^{24} \) | |
| 47 | \( 1 - 15340 T^{2} + 115278210 T^{4} - 568110772796 T^{6} + 2081045592203375 T^{8} - 6074523141261659352 T^{10} + \)\(14\!\cdots\!72\)\( T^{12} - 6074523141261659352 p^{4} T^{14} + 2081045592203375 p^{8} T^{16} - 568110772796 p^{12} T^{18} + 115278210 p^{16} T^{20} - 15340 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 52 T + 9298 T^{2} + 421060 T^{3} + 49987391 T^{4} + 1862295400 T^{5} + 167186937020 T^{6} + 1862295400 p^{2} T^{7} + 49987391 p^{4} T^{8} + 421060 p^{6} T^{9} + 9298 p^{8} T^{10} + 52 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 24044 T^{2} + 299852450 T^{4} - 2509844879740 T^{6} + 15610492757496335 T^{8} - 75760589297731471320 T^{10} + \)\(29\!\cdots\!32\)\( T^{12} - 75760589297731471320 p^{4} T^{14} + 15610492757496335 p^{8} T^{16} - 2509844879740 p^{12} T^{18} + 299852450 p^{16} T^{20} - 24044 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 - 52 T + 12018 T^{2} - 648452 T^{3} + 77228831 T^{4} - 3551171688 T^{5} + 345023520508 T^{6} - 3551171688 p^{2} T^{7} + 77228831 p^{4} T^{8} - 648452 p^{6} T^{9} + 12018 p^{8} T^{10} - 52 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 - 35916 T^{2} + 630632098 T^{4} - 7183251116764 T^{6} + 59381050864764239 T^{8} - \)\(37\!\cdots\!32\)\( T^{10} + \)\(18\!\cdots\!24\)\( T^{12} - \)\(37\!\cdots\!32\)\( p^{4} T^{14} + 59381050864764239 p^{8} T^{16} - 7183251116764 p^{12} T^{18} + 630632098 p^{16} T^{20} - 35916 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( 1 - 6116 T^{2} + 91060386 T^{4} - 373985966164 T^{6} + 3852234631502255 T^{8} - 12292034220711508296 T^{10} + \)\(11\!\cdots\!08\)\( T^{12} - 12292034220711508296 p^{4} T^{14} + 3852234631502255 p^{8} T^{16} - 373985966164 p^{12} T^{18} + 91060386 p^{16} T^{20} - 6116 p^{20} T^{22} + p^{24} T^{24} \) | |
| 73 | \( ( 1 - 156 T + 24658 T^{2} - 2501996 T^{3} + 241574543 T^{4} - 18743454904 T^{5} + 1488139178108 T^{6} - 18743454904 p^{2} T^{7} + 241574543 p^{4} T^{8} - 2501996 p^{6} T^{9} + 24658 p^{8} T^{10} - 156 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 - 45804 T^{2} + 1023374466 T^{4} - 15014631100220 T^{6} + 163282873474457967 T^{8} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(96\!\cdots\!84\)\( T^{12} - \)\(13\!\cdots\!40\)\( p^{4} T^{14} + 163282873474457967 p^{8} T^{16} - 15014631100220 p^{12} T^{18} + 1023374466 p^{16} T^{20} - 45804 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( 1 - 64972 T^{2} + 2024986146 T^{4} - 39980120746844 T^{6} + 556490075320955855 T^{8} - \)\(57\!\cdots\!12\)\( T^{10} + \)\(45\!\cdots\!60\)\( T^{12} - \)\(57\!\cdots\!12\)\( p^{4} T^{14} + 556490075320955855 p^{8} T^{16} - 39980120746844 p^{12} T^{18} + 2024986146 p^{16} T^{20} - 64972 p^{20} T^{22} + p^{24} T^{24} \) | |
| 89 | \( ( 1 - 276 T + 44854 T^{2} - 4852292 T^{3} + 439252367 T^{4} - 35194402216 T^{5} + 3057470352308 T^{6} - 35194402216 p^{2} T^{7} + 439252367 p^{4} T^{8} - 4852292 p^{6} T^{9} + 44854 p^{8} T^{10} - 276 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 132 T + 27762 T^{2} + 3615988 T^{3} + 481229487 T^{4} + 47786875272 T^{5} + 5516027091132 T^{6} + 47786875272 p^{2} T^{7} + 481229487 p^{4} T^{8} + 3615988 p^{6} T^{9} + 27762 p^{8} T^{10} + 132 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
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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
−3.82000673284375465430718180731, −3.81445874117296093359958317657, −3.75754049898247980980210808788, −3.65176569466743874605183804283, −3.62435914463299522053350687351, −3.20733304763936678510245947722, −3.15909408335681994726978260989, −3.13656635395539897917357889250, −2.81065231572611106103232995187, −2.69715197689494351950952293759, −2.60103581964061693665602016701, −2.59131714489194281174877390185, −2.36108942259266801391497127990, −2.13884848421528474479242830432, −2.08014835690906896639222213825, −1.88455686670834941379534769399, −1.72766111432829072856611742054, −1.63852464079462922224859826582, −1.57238591610922448635521579800, −1.15291327862692365517215594656, −0.936449662118808982843596533873, −0.826442319181972742388795644393, −0.51180188813846074168325141557, −0.31952249809876285105858679373, −0.23811325842966210425557025537, 0.23811325842966210425557025537, 0.31952249809876285105858679373, 0.51180188813846074168325141557, 0.826442319181972742388795644393, 0.936449662118808982843596533873, 1.15291327862692365517215594656, 1.57238591610922448635521579800, 1.63852464079462922224859826582, 1.72766111432829072856611742054, 1.88455686670834941379534769399, 2.08014835690906896639222213825, 2.13884848421528474479242830432, 2.36108942259266801391497127990, 2.59131714489194281174877390185, 2.60103581964061693665602016701, 2.69715197689494351950952293759, 2.81065231572611106103232995187, 3.13656635395539897917357889250, 3.15909408335681994726978260989, 3.20733304763936678510245947722, 3.62435914463299522053350687351, 3.65176569466743874605183804283, 3.75754049898247980980210808788, 3.81445874117296093359958317657, 3.82000673284375465430718180731
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.