Dirichlet series
| L(s) = 1 | + 714·7-s − 5.94e4·19-s − 1.48e4·25-s + 2.19e5·31-s − 1.66e5·37-s − 3.26e5·43-s + 1.88e5·49-s + 7.36e5·61-s + 1.76e5·67-s − 2.93e6·73-s + 4.49e5·79-s − 1.72e6·103-s − 1.18e6·109-s + 1.58e6·121-s + 127-s + 131-s − 4.24e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.05e7·169-s + 173-s − 1.05e7·175-s + ⋯ |
| L(s) = 1 | + 2.08·7-s − 8.66·19-s − 0.949·25-s + 7.37·31-s − 3.28·37-s − 4.11·43-s + 1.60·49-s + 3.24·61-s + 0.585·67-s − 7.53·73-s + 0.911·79-s − 1.57·103-s − 0.915·109-s + 0.895·121-s − 18.0·133-s + 4.26·169-s − 1.97·175-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44133\times 10^{21}\) |
| Root analytic conductor: | \(7.61404\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [3]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(3.104533182\) |
| \(L(\frac12)\) | \(\approx\) | \(3.104533182\) |
| \(L(4)\) | 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 | \( 1 \) | |
| 7 | \( ( 1 - 51 p T + 282 p^{3} T^{2} - 3127 p^{5} T^{3} + 282 p^{9} T^{4} - 51 p^{13} T^{5} + p^{18} T^{6} )^{2} \) | |
| good | 5 | \( 1 + 14832 T^{2} - 471007656 T^{4} - 595335302468 p T^{6} + 8427785157183096 p^{2} T^{8} + 718602944463024768 p^{4} T^{10} - \)\(36\!\cdots\!26\)\( p^{6} T^{12} + 718602944463024768 p^{16} T^{14} + 8427785157183096 p^{26} T^{16} - 595335302468 p^{37} T^{18} - 471007656 p^{48} T^{20} + 14832 p^{60} T^{22} + p^{72} T^{24} \) |
| 11 | \( 1 - 1587228 T^{2} + 3165393347760 T^{4} - 17411600271807526604 T^{6} + \)\(17\!\cdots\!20\)\( T^{8} - \)\(28\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!38\)\( T^{12} - \)\(28\!\cdots\!40\)\( p^{12} T^{14} + \)\(17\!\cdots\!20\)\( p^{24} T^{16} - 17411600271807526604 p^{36} T^{18} + 3165393347760 p^{48} T^{20} - 1587228 p^{60} T^{22} + p^{72} T^{24} \) | |
| 13 | \( ( 1 - 10285083 T^{2} + 86801488569963 T^{4} - \)\(43\!\cdots\!46\)\( T^{6} + 86801488569963 p^{12} T^{8} - 10285083 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 17 | \( 1 + 27210798 T^{2} - 4945912639227 p T^{4} - \)\(48\!\cdots\!38\)\( T^{6} - \)\(76\!\cdots\!62\)\( T^{8} + \)\(12\!\cdots\!58\)\( T^{10} + \)\(99\!\cdots\!05\)\( T^{12} + \)\(12\!\cdots\!58\)\( p^{12} T^{14} - \)\(76\!\cdots\!62\)\( p^{24} T^{16} - \)\(48\!\cdots\!38\)\( p^{36} T^{18} - 4945912639227 p^{49} T^{20} + 27210798 p^{60} T^{22} + p^{72} T^{24} \) | |
| 19 | \( ( 1 + 29733 T + 515342670 T^{2} + 6560851281831 T^{3} + 67127992122580428 T^{4} + \)\(57\!\cdots\!89\)\( T^{5} + \)\(42\!\cdots\!76\)\( T^{6} + \)\(57\!\cdots\!89\)\( p^{6} T^{7} + 67127992122580428 p^{12} T^{8} + 6560851281831 p^{18} T^{9} + 515342670 p^{24} T^{10} + 29733 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 23 | \( 1 - 378724758 T^{2} + 51565349524009125 T^{4} - \)\(50\!\cdots\!74\)\( T^{6} + \)\(79\!\cdots\!50\)\( T^{8} - \)\(22\!\cdots\!90\)\( T^{10} - \)\(11\!\cdots\!07\)\( T^{12} - \)\(22\!\cdots\!90\)\( p^{12} T^{14} + \)\(79\!\cdots\!50\)\( p^{24} T^{16} - \)\(50\!\cdots\!74\)\( p^{36} T^{18} + 51565349524009125 p^{48} T^{20} - 378724758 p^{60} T^{22} + p^{72} T^{24} \) | |
| 29 | \( ( 1 + 3129830556 T^{2} + 4318273516519659264 T^{4} + \)\(33\!\cdots\!38\)\( T^{6} + 4318273516519659264 p^{12} T^{8} + 3129830556 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 109917 T + 7211026371 T^{2} - 349951261355136 T^{3} + 13653202607573683389 T^{4} - \)\(46\!\cdots\!19\)\( T^{5} + \)\(14\!\cdots\!42\)\( T^{6} - \)\(46\!\cdots\!19\)\( p^{6} T^{7} + 13653202607573683389 p^{12} T^{8} - 349951261355136 p^{18} T^{9} + 7211026371 p^{24} T^{10} - 109917 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 37 | \( ( 1 + 83121 T - 2428719402 T^{2} - 75165326989385 T^{3} + 27854713873978597188 T^{4} + \)\(53\!\cdots\!89\)\( T^{5} - \)\(54\!\cdots\!68\)\( T^{6} + \)\(53\!\cdots\!89\)\( p^{6} T^{7} + 27854713873978597188 p^{12} T^{8} - 75165326989385 p^{18} T^{9} - 2428719402 p^{24} T^{10} + 83121 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 18832053582 T^{2} + \)\(17\!\cdots\!23\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{6} + \)\(17\!\cdots\!23\)\( p^{12} T^{8} - 18832053582 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 81729 T + 9079346835 T^{2} + 964777882825442 T^{3} + 9079346835 p^{6} T^{4} + 81729 p^{12} T^{5} + p^{18} T^{6} )^{4} \) | |
| 47 | \( 1 - 7357085226 T^{2} + 82753628094104385285 T^{4} - \)\(27\!\cdots\!62\)\( T^{6} + \)\(15\!\cdots\!98\)\( T^{8} - \)\(19\!\cdots\!26\)\( T^{10} + \)\(45\!\cdots\!69\)\( T^{12} - \)\(19\!\cdots\!26\)\( p^{12} T^{14} + \)\(15\!\cdots\!98\)\( p^{24} T^{16} - \)\(27\!\cdots\!62\)\( p^{36} T^{18} + 82753628094104385285 p^{48} T^{20} - 7357085226 p^{60} T^{22} + p^{72} T^{24} \) | |
| 53 | \( 1 - 92240514708 T^{2} + \)\(42\!\cdots\!60\)\( T^{4} - \)\(15\!\cdots\!24\)\( T^{6} + \)\(49\!\cdots\!20\)\( T^{8} - \)\(13\!\cdots\!40\)\( T^{10} + \)\(31\!\cdots\!18\)\( T^{12} - \)\(13\!\cdots\!40\)\( p^{12} T^{14} + \)\(49\!\cdots\!20\)\( p^{24} T^{16} - \)\(15\!\cdots\!24\)\( p^{36} T^{18} + \)\(42\!\cdots\!60\)\( p^{48} T^{20} - 92240514708 p^{60} T^{22} + p^{72} T^{24} \) | |
| 59 | \( 1 + 56135697120 T^{2} - \)\(24\!\cdots\!00\)\( T^{4} - \)\(54\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!60\)\( T^{8} + \)\(97\!\cdots\!20\)\( T^{10} - \)\(21\!\cdots\!74\)\( T^{12} + \)\(97\!\cdots\!20\)\( p^{12} T^{14} + \)\(11\!\cdots\!60\)\( p^{24} T^{16} - \)\(54\!\cdots\!24\)\( p^{36} T^{18} - \)\(24\!\cdots\!00\)\( p^{48} T^{20} + 56135697120 p^{60} T^{22} + p^{72} T^{24} \) | |
| 61 | \( ( 1 - 368004 T + 124227857127 T^{2} - 29103795965609820 T^{3} + \)\(52\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} - \)\(10\!\cdots\!96\)\( p^{6} T^{7} + \)\(52\!\cdots\!38\)\( p^{12} T^{8} - 29103795965609820 p^{18} T^{9} + 124227857127 p^{24} T^{10} - 368004 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 67 | \( ( 1 - 88119 T - 229159605546 T^{2} + 3920579216228111 T^{3} + \)\(33\!\cdots\!48\)\( T^{4} + \)\(11\!\cdots\!41\)\( T^{5} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!41\)\( p^{6} T^{7} + \)\(33\!\cdots\!48\)\( p^{12} T^{8} + 3920579216228111 p^{18} T^{9} - 229159605546 p^{24} T^{10} - 88119 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 477903790230 T^{2} + \)\(10\!\cdots\!71\)\( T^{4} + \)\(15\!\cdots\!44\)\( T^{6} + \)\(10\!\cdots\!71\)\( p^{12} T^{8} + 477903790230 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 1465509 T + 1276427859486 T^{2} + 821450499471512631 T^{3} + \)\(41\!\cdots\!20\)\( T^{4} + \)\(18\!\cdots\!81\)\( T^{5} + \)\(74\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!81\)\( p^{6} T^{7} + \)\(41\!\cdots\!20\)\( p^{12} T^{8} + 821450499471512631 p^{18} T^{9} + 1276427859486 p^{24} T^{10} + 1465509 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 224745 T - 258938366817 T^{2} + 304137163150749680 T^{3} - \)\(21\!\cdots\!43\)\( T^{4} - \)\(37\!\cdots\!35\)\( T^{5} + \)\(37\!\cdots\!78\)\( T^{6} - \)\(37\!\cdots\!35\)\( p^{6} T^{7} - \)\(21\!\cdots\!43\)\( p^{12} T^{8} + 304137163150749680 p^{18} T^{9} - 258938366817 p^{24} T^{10} - 224745 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 83 | \( ( 1 - 17718458976 p T^{2} + \)\(10\!\cdots\!88\)\( T^{4} - \)\(42\!\cdots\!26\)\( T^{6} + \)\(10\!\cdots\!88\)\( p^{12} T^{8} - 17718458976 p^{25} T^{10} + p^{36} T^{12} )^{2} \) | |
| 89 | \( 1 + 1121758631790 T^{2} + \)\(98\!\cdots\!65\)\( T^{4} + \)\(40\!\cdots\!66\)\( T^{6} + \)\(62\!\cdots\!10\)\( T^{8} - \)\(94\!\cdots\!90\)\( T^{10} - \)\(62\!\cdots\!39\)\( T^{12} - \)\(94\!\cdots\!90\)\( p^{12} T^{14} + \)\(62\!\cdots\!10\)\( p^{24} T^{16} + \)\(40\!\cdots\!66\)\( p^{36} T^{18} + \)\(98\!\cdots\!65\)\( p^{48} T^{20} + 1121758631790 p^{60} T^{22} + p^{72} T^{24} \) | |
| 97 | \( ( 1 - 4108600106736 T^{2} + \)\(75\!\cdots\!88\)\( T^{4} - \)\(79\!\cdots\!02\)\( T^{6} + \)\(75\!\cdots\!88\)\( p^{12} T^{8} - 4108600106736 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
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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
−2.92632460632822886974500144381, −2.91690886537783415248927032933, −2.62985714986477367844236365906, −2.60061931116444993243446735422, −2.48806473834204228060517955314, −2.47613611568915082507427840827, −2.19443108830856731759527251919, −2.07213096832251154513847779516, −2.02835503655202556259756874933, −1.90199920553713270066658071936, −1.78132736176190264661380305641, −1.76372033732659868789775890158, −1.61790826060891240419497301101, −1.61501841155178359219836598502, −1.47856385747897687920666120039, −1.28192754698812816453954857197, −1.08159161075793925577308554552, −1.02420878338621933668528901613, −0.72916496013807189308019609397, −0.71120885428297843201953644608, −0.48703064914161974980776239056, −0.43133634796558311069974433156, −0.26588141388395984192878893323, −0.17142214737822785945714467660, −0.12474717736204405970224020915, 0.12474717736204405970224020915, 0.17142214737822785945714467660, 0.26588141388395984192878893323, 0.43133634796558311069974433156, 0.48703064914161974980776239056, 0.71120885428297843201953644608, 0.72916496013807189308019609397, 1.02420878338621933668528901613, 1.08159161075793925577308554552, 1.28192754698812816453954857197, 1.47856385747897687920666120039, 1.61501841155178359219836598502, 1.61790826060891240419497301101, 1.76372033732659868789775890158, 1.78132736176190264661380305641, 1.90199920553713270066658071936, 2.02835503655202556259756874933, 2.07213096832251154513847779516, 2.19443108830856731759527251919, 2.47613611568915082507427840827, 2.48806473834204228060517955314, 2.60061931116444993243446735422, 2.62985714986477367844236365906, 2.91690886537783415248927032933, 2.92632460632822886974500144381
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.