Dirichlet series
| L(s) = 1 | − 952·3-s + 2.56e5·5-s − 4.94e6·7-s − 1.29e7·9-s − 1.39e8·11-s + 3.10e8·13-s − 2.44e8·15-s + 1.37e9·17-s + 3.44e9·19-s + 4.70e9·21-s + 7.02e9·23-s − 3.55e10·25-s + 1.49e10·27-s + 6.87e10·29-s + 3.27e10·31-s + 1.32e11·33-s − 1.26e12·35-s − 2.73e11·37-s − 2.95e11·39-s − 3.58e11·41-s − 8.58e11·43-s − 3.32e12·45-s + 3.13e12·47-s + 1.42e13·49-s − 1.30e12·51-s + 1.26e13·53-s − 3.58e13·55-s + ⋯ |
| L(s) = 1 | − 0.251·3-s + 1.46·5-s − 2.26·7-s − 0.903·9-s − 2.15·11-s + 1.37·13-s − 0.369·15-s + 0.810·17-s + 0.884·19-s + 0.569·21-s + 0.430·23-s − 1.16·25-s + 0.274·27-s + 0.740·29-s + 0.213·31-s + 0.542·33-s − 3.33·35-s − 0.473·37-s − 0.344·39-s − 0.287·41-s − 0.481·43-s − 1.32·45-s + 0.903·47-s + 3·49-s − 0.203·51-s + 1.47·53-s − 3.17·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(16-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.66622\times 10^{13}\) |
| Root analytic conductor: | \(12.6418\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 7^{6} ,\ ( \ : [15/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(8)\) | \(\approx\) | \(2.818190415\) |
| \(L(\frac12)\) | \(\approx\) | \(2.818190415\) |
| \(L(\frac{17}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p^{7} T )^{6} \) | |
| good | 3 | \( 1 + 952 T + 13869898 T^{2} + 1179009160 p^{2} T^{3} + 27410549728079 p^{2} T^{4} + 755614583323600 p^{5} T^{5} + 985632771129170828 p^{8} T^{6} + 755614583323600 p^{20} T^{7} + 27410549728079 p^{32} T^{8} + 1179009160 p^{47} T^{9} + 13869898 p^{60} T^{10} + 952 p^{75} T^{11} + p^{90} T^{12} \) |
| 5 | \( 1 - 256788 T + 4060064466 p^{2} T^{2} - 957939446356564 p^{2} T^{3} + 46182106749342744387 p^{3} T^{4} - \)\(77\!\cdots\!56\)\( p^{6} T^{5} + \)\(67\!\cdots\!84\)\( p^{5} T^{6} - \)\(77\!\cdots\!56\)\( p^{21} T^{7} + 46182106749342744387 p^{33} T^{8} - 957939446356564 p^{47} T^{9} + 4060064466 p^{62} T^{10} - 256788 p^{75} T^{11} + p^{90} T^{12} \) | |
| 11 | \( 1 + 139572168 T + 17419800036663138 T^{2} + \)\(13\!\cdots\!64\)\( p T^{3} + \)\(10\!\cdots\!31\)\( p^{2} T^{4} + \)\(76\!\cdots\!16\)\( p^{3} T^{5} + \)\(48\!\cdots\!88\)\( p^{4} T^{6} + \)\(76\!\cdots\!16\)\( p^{18} T^{7} + \)\(10\!\cdots\!31\)\( p^{32} T^{8} + \)\(13\!\cdots\!64\)\( p^{46} T^{9} + 17419800036663138 p^{60} T^{10} + 139572168 p^{75} T^{11} + p^{90} T^{12} \) | |
| 13 | \( 1 - 310264836 T + 13619490639247098 p T^{2} - \)\(30\!\cdots\!88\)\( p^{2} T^{3} + \)\(77\!\cdots\!99\)\( p^{3} T^{4} - \)\(14\!\cdots\!68\)\( p^{4} T^{5} + \)\(29\!\cdots\!44\)\( p^{5} T^{6} - \)\(14\!\cdots\!68\)\( p^{19} T^{7} + \)\(77\!\cdots\!99\)\( p^{33} T^{8} - \)\(30\!\cdots\!88\)\( p^{47} T^{9} + 13619490639247098 p^{61} T^{10} - 310264836 p^{75} T^{11} + p^{90} T^{12} \) | |
| 17 | \( 1 - 1371963180 T + 6346645925638984194 T^{2} - \)\(56\!\cdots\!44\)\( p T^{3} + \)\(28\!\cdots\!19\)\( T^{4} - \)\(32\!\cdots\!68\)\( T^{5} + \)\(92\!\cdots\!16\)\( T^{6} - \)\(32\!\cdots\!68\)\( p^{15} T^{7} + \)\(28\!\cdots\!19\)\( p^{30} T^{8} - \)\(56\!\cdots\!44\)\( p^{46} T^{9} + 6346645925638984194 p^{60} T^{10} - 1371963180 p^{75} T^{11} + p^{90} T^{12} \) | |
| 19 | \( 1 - 3446059176 T + 60966466812720039018 T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(16\!\cdots\!83\)\( T^{4} - \)\(21\!\cdots\!36\)\( T^{5} + \)\(28\!\cdots\!72\)\( T^{6} - \)\(21\!\cdots\!36\)\( p^{15} T^{7} + \)\(16\!\cdots\!83\)\( p^{30} T^{8} - \)\(12\!\cdots\!92\)\( p^{45} T^{9} + 60966466812720039018 p^{60} T^{10} - 3446059176 p^{75} T^{11} + p^{90} T^{12} \) | |
| 23 | \( 1 - 7026527616 T + \)\(55\!\cdots\!78\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!43\)\( T^{4} - \)\(70\!\cdots\!88\)\( T^{5} + \)\(41\!\cdots\!56\)\( T^{6} - \)\(70\!\cdots\!88\)\( p^{15} T^{7} + \)\(14\!\cdots\!43\)\( p^{30} T^{8} - \)\(13\!\cdots\!40\)\( p^{45} T^{9} + \)\(55\!\cdots\!78\)\( p^{60} T^{10} - 7026527616 p^{75} T^{11} + p^{90} T^{12} \) | |
| 29 | \( 1 - 68767761540 T + \)\(37\!\cdots\!10\)\( T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(65\!\cdots\!99\)\( T^{4} - \)\(43\!\cdots\!68\)\( T^{5} + \)\(70\!\cdots\!92\)\( T^{6} - \)\(43\!\cdots\!68\)\( p^{15} T^{7} + \)\(65\!\cdots\!99\)\( p^{30} T^{8} - \)\(26\!\cdots\!80\)\( p^{45} T^{9} + \)\(37\!\cdots\!10\)\( p^{60} T^{10} - 68767761540 p^{75} T^{11} + p^{90} T^{12} \) | |
| 31 | \( 1 - 32780361600 T + \)\(19\!\cdots\!02\)\( p T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!23\)\( T^{4} + \)\(58\!\cdots\!80\)\( T^{5} + \)\(49\!\cdots\!24\)\( T^{6} + \)\(58\!\cdots\!80\)\( p^{15} T^{7} + \)\(19\!\cdots\!23\)\( p^{30} T^{8} + \)\(74\!\cdots\!80\)\( p^{45} T^{9} + \)\(19\!\cdots\!02\)\( p^{61} T^{10} - 32780361600 p^{75} T^{11} + p^{90} T^{12} \) | |
| 37 | \( 1 + 273369767628 T + \)\(13\!\cdots\!98\)\( T^{2} + \)\(59\!\cdots\!80\)\( T^{3} + \)\(76\!\cdots\!15\)\( T^{4} + \)\(44\!\cdots\!72\)\( T^{5} + \)\(28\!\cdots\!96\)\( T^{6} + \)\(44\!\cdots\!72\)\( p^{15} T^{7} + \)\(76\!\cdots\!15\)\( p^{30} T^{8} + \)\(59\!\cdots\!80\)\( p^{45} T^{9} + \)\(13\!\cdots\!98\)\( p^{60} T^{10} + 273369767628 p^{75} T^{11} + p^{90} T^{12} \) | |
| 41 | \( 1 + 358650707268 T + \)\(82\!\cdots\!10\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(29\!\cdots\!95\)\( T^{4} + \)\(79\!\cdots\!36\)\( T^{5} + \)\(60\!\cdots\!08\)\( T^{6} + \)\(79\!\cdots\!36\)\( p^{15} T^{7} + \)\(29\!\cdots\!95\)\( p^{30} T^{8} + \)\(26\!\cdots\!56\)\( p^{45} T^{9} + \)\(82\!\cdots\!10\)\( p^{60} T^{10} + 358650707268 p^{75} T^{11} + p^{90} T^{12} \) | |
| 43 | \( 1 + 858885323496 T + \)\(15\!\cdots\!10\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!87\)\( T^{4} + \)\(83\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!60\)\( p T^{6} + \)\(83\!\cdots\!24\)\( p^{15} T^{7} + \)\(10\!\cdots\!87\)\( p^{30} T^{8} + \)\(13\!\cdots\!60\)\( p^{45} T^{9} + \)\(15\!\cdots\!10\)\( p^{60} T^{10} + 858885323496 p^{75} T^{11} + p^{90} T^{12} \) | |
| 47 | \( 1 - 3139234928256 T + \)\(82\!\cdots\!02\)\( T^{2} - \)\(11\!\cdots\!08\)\( p T^{3} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(70\!\cdots\!88\)\( T^{5} + \)\(48\!\cdots\!00\)\( T^{6} - \)\(70\!\cdots\!88\)\( p^{15} T^{7} + \)\(14\!\cdots\!15\)\( p^{30} T^{8} - \)\(11\!\cdots\!08\)\( p^{46} T^{9} + \)\(82\!\cdots\!02\)\( p^{60} T^{10} - 3139234928256 p^{75} T^{11} + p^{90} T^{12} \) | |
| 53 | \( 1 - 12619016040276 T + \)\(25\!\cdots\!62\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!83\)\( T^{4} - \)\(23\!\cdots\!12\)\( T^{5} + \)\(25\!\cdots\!20\)\( T^{6} - \)\(23\!\cdots\!12\)\( p^{15} T^{7} + \)\(32\!\cdots\!83\)\( p^{30} T^{8} - \)\(23\!\cdots\!40\)\( p^{45} T^{9} + \)\(25\!\cdots\!62\)\( p^{60} T^{10} - 12619016040276 p^{75} T^{11} + p^{90} T^{12} \) | |
| 59 | \( 1 - 4557388553592 T + \)\(12\!\cdots\!74\)\( T^{2} - \)\(50\!\cdots\!72\)\( T^{3} + \)\(74\!\cdots\!67\)\( T^{4} - \)\(64\!\cdots\!60\)\( p T^{5} + \)\(31\!\cdots\!52\)\( T^{6} - \)\(64\!\cdots\!60\)\( p^{16} T^{7} + \)\(74\!\cdots\!67\)\( p^{30} T^{8} - \)\(50\!\cdots\!72\)\( p^{45} T^{9} + \)\(12\!\cdots\!74\)\( p^{60} T^{10} - 4557388553592 p^{75} T^{11} + p^{90} T^{12} \) | |
| 61 | \( 1 - 22358666996484 T + \)\(66\!\cdots\!74\)\( T^{2} + \)\(89\!\cdots\!44\)\( T^{3} + \)\(72\!\cdots\!87\)\( T^{4} - \)\(16\!\cdots\!92\)\( T^{5} + \)\(45\!\cdots\!08\)\( T^{6} - \)\(16\!\cdots\!92\)\( p^{15} T^{7} + \)\(72\!\cdots\!87\)\( p^{30} T^{8} + \)\(89\!\cdots\!44\)\( p^{45} T^{9} + \)\(66\!\cdots\!74\)\( p^{60} T^{10} - 22358666996484 p^{75} T^{11} + p^{90} T^{12} \) | |
| 67 | \( 1 + 55386392259384 T + \)\(14\!\cdots\!54\)\( T^{2} + \)\(66\!\cdots\!36\)\( T^{3} + \)\(91\!\cdots\!91\)\( T^{4} + \)\(32\!\cdots\!56\)\( T^{5} + \)\(29\!\cdots\!12\)\( T^{6} + \)\(32\!\cdots\!56\)\( p^{15} T^{7} + \)\(91\!\cdots\!91\)\( p^{30} T^{8} + \)\(66\!\cdots\!36\)\( p^{45} T^{9} + \)\(14\!\cdots\!54\)\( p^{60} T^{10} + 55386392259384 p^{75} T^{11} + p^{90} T^{12} \) | |
| 71 | \( 1 + 215558440144368 T + \)\(30\!\cdots\!22\)\( T^{2} + \)\(42\!\cdots\!92\)\( T^{3} + \)\(43\!\cdots\!87\)\( T^{4} + \)\(39\!\cdots\!96\)\( T^{5} + \)\(33\!\cdots\!68\)\( T^{6} + \)\(39\!\cdots\!96\)\( p^{15} T^{7} + \)\(43\!\cdots\!87\)\( p^{30} T^{8} + \)\(42\!\cdots\!92\)\( p^{45} T^{9} + \)\(30\!\cdots\!22\)\( p^{60} T^{10} + 215558440144368 p^{75} T^{11} + p^{90} T^{12} \) | |
| 73 | \( 1 + 227606379352548 T + \)\(42\!\cdots\!34\)\( T^{2} + \)\(55\!\cdots\!84\)\( T^{3} + \)\(58\!\cdots\!79\)\( T^{4} + \)\(56\!\cdots\!28\)\( T^{5} + \)\(51\!\cdots\!40\)\( T^{6} + \)\(56\!\cdots\!28\)\( p^{15} T^{7} + \)\(58\!\cdots\!79\)\( p^{30} T^{8} + \)\(55\!\cdots\!84\)\( p^{45} T^{9} + \)\(42\!\cdots\!34\)\( p^{60} T^{10} + 227606379352548 p^{75} T^{11} + p^{90} T^{12} \) | |
| 79 | \( 1 + 534756733862880 T + \)\(18\!\cdots\!42\)\( T^{2} + \)\(50\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!59\)\( T^{4} + \)\(19\!\cdots\!24\)\( T^{5} + \)\(34\!\cdots\!24\)\( T^{6} + \)\(19\!\cdots\!24\)\( p^{15} T^{7} + \)\(10\!\cdots\!59\)\( p^{30} T^{8} + \)\(50\!\cdots\!92\)\( p^{45} T^{9} + \)\(18\!\cdots\!42\)\( p^{60} T^{10} + 534756733862880 p^{75} T^{11} + p^{90} T^{12} \) | |
| 83 | \( 1 + 32818220604120 T + \)\(67\!\cdots\!50\)\( T^{2} - \)\(86\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!59\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!88\)\( T^{6} + \)\(38\!\cdots\!20\)\( p^{15} T^{7} + \)\(65\!\cdots\!59\)\( p^{30} T^{8} - \)\(86\!\cdots\!00\)\( p^{45} T^{9} + \)\(67\!\cdots\!50\)\( p^{60} T^{10} + 32818220604120 p^{75} T^{11} + p^{90} T^{12} \) | |
| 89 | \( 1 - 1904714355388572 T + \)\(23\!\cdots\!82\)\( T^{2} - \)\(19\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!55\)\( T^{4} - \)\(73\!\cdots\!44\)\( T^{5} + \)\(33\!\cdots\!08\)\( T^{6} - \)\(73\!\cdots\!44\)\( p^{15} T^{7} + \)\(13\!\cdots\!55\)\( p^{30} T^{8} - \)\(19\!\cdots\!56\)\( p^{45} T^{9} + \)\(23\!\cdots\!82\)\( p^{60} T^{10} - 1904714355388572 p^{75} T^{11} + p^{90} T^{12} \) | |
| 97 | \( 1 - 425879188281708 T + \)\(17\!\cdots\!62\)\( T^{2} - \)\(62\!\cdots\!28\)\( T^{3} + \)\(17\!\cdots\!99\)\( T^{4} - \)\(51\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!20\)\( T^{6} - \)\(51\!\cdots\!92\)\( p^{15} T^{7} + \)\(17\!\cdots\!99\)\( p^{30} T^{8} - \)\(62\!\cdots\!28\)\( p^{45} T^{9} + \)\(17\!\cdots\!62\)\( p^{60} T^{10} - 425879188281708 p^{75} T^{11} + p^{90} T^{12} \) | |
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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
−5.11288639186382951719325563628, −4.72113801927512917483230766042, −4.47408178548621176601260306679, −4.44921475144359944259634815456, −4.03482076392510731645532549450, −3.99175939276262993860719027670, −3.74516652807233797302932210497, −3.29887685102128980101855965557, −3.19861376756133661837846766731, −3.12482291963682851579709041287, −2.98364219974969958286743242046, −2.95390101691023444661848199139, −2.64610993161438332250186816416, −2.29806706221042255979563670136, −2.21047260912819157283787048628, −1.98225228756650815110138473064, −1.86520392868447971770553369015, −1.48076427819795794394173747274, −1.37413560681268288792408340019, −1.09064628771447449872081580432, −0.913793635342087931696508386791, −0.56866589366767846290621882956, −0.51226414633475546399080119644, −0.30164296267835885986087268759, −0.15859165831346925518174478873, 0.15859165831346925518174478873, 0.30164296267835885986087268759, 0.51226414633475546399080119644, 0.56866589366767846290621882956, 0.913793635342087931696508386791, 1.09064628771447449872081580432, 1.37413560681268288792408340019, 1.48076427819795794394173747274, 1.86520392868447971770553369015, 1.98225228756650815110138473064, 2.21047260912819157283787048628, 2.29806706221042255979563670136, 2.64610993161438332250186816416, 2.95390101691023444661848199139, 2.98364219974969958286743242046, 3.12482291963682851579709041287, 3.19861376756133661837846766731, 3.29887685102128980101855965557, 3.74516652807233797302932210497, 3.99175939276262993860719027670, 4.03482076392510731645532549450, 4.44921475144359944259634815456, 4.47408178548621176601260306679, 4.72113801927512917483230766042, 5.11288639186382951719325563628
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.