Dirichlet series
| L(s) = 1 | + 15·4-s − 14·5-s − 63·9-s − 154·11-s + 39·16-s − 52·19-s − 210·20-s − 85·25-s − 1.14e3·29-s − 280·31-s − 945·36-s + 1.79e3·41-s − 2.31e3·44-s + 882·45-s + 2.10e3·49-s + 2.15e3·55-s − 2.63e3·59-s − 772·61-s − 347·64-s + 1.60e3·71-s − 780·76-s + 748·79-s − 546·80-s + 2.26e3·81-s − 1.38e3·89-s + 728·95-s + 9.70e3·99-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 1.25·5-s − 7/3·9-s − 4.22·11-s + 0.609·16-s − 0.627·19-s − 2.34·20-s − 0.679·25-s − 7.32·29-s − 1.62·31-s − 4.37·36-s + 6.82·41-s − 7.91·44-s + 2.92·45-s + 6.13·49-s + 5.28·55-s − 5.80·59-s − 1.62·61-s − 0.677·64-s + 2.68·71-s − 1.17·76-s + 1.06·79-s − 0.763·80-s + 28/9·81-s − 1.65·89-s + 0.786·95-s + 9.84·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{14} \cdot 11^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 5^{14} \cdot 11^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{14} \cdot 5^{14} \cdot 11^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.86910\times 10^{13}\) |
| Root analytic conductor: | \(3.12014\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{14} \cdot 5^{14} \cdot 11^{14} ,\ ( \ : [3/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.04949157958\) |
| \(L(\frac12)\) | \(\approx\) | \(0.04949157958\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{7} \) |
| 5 | \( 1 + 14 T + 281 T^{2} + 2344 T^{3} + 32571 T^{4} + 81034 p T^{5} + 252067 p^{2} T^{6} + 627856 p^{3} T^{7} + 252067 p^{5} T^{8} + 81034 p^{7} T^{9} + 32571 p^{9} T^{10} + 2344 p^{12} T^{11} + 281 p^{15} T^{12} + 14 p^{18} T^{13} + p^{21} T^{14} \) | |
| 11 | \( ( 1 + p T )^{14} \) | |
| good | 2 | \( 1 - 15 T^{2} + 93 p T^{4} - 929 p T^{6} + 14813 T^{8} - 76871 T^{10} + 28373 p^{4} T^{12} - 19577 p^{7} T^{14} + 28373 p^{10} T^{16} - 76871 p^{12} T^{18} + 14813 p^{18} T^{20} - 929 p^{25} T^{22} + 93 p^{31} T^{24} - 15 p^{36} T^{26} + p^{42} T^{28} \) |
| 7 | \( 1 - 2106 T^{2} + 2542063 T^{4} - 2176671804 T^{6} + 1444827037981 T^{8} - 776415699323526 T^{10} + 345946604474390035 T^{12} - \)\(12\!\cdots\!28\)\( T^{14} + 345946604474390035 p^{6} T^{16} - 776415699323526 p^{12} T^{18} + 1444827037981 p^{18} T^{20} - 2176671804 p^{24} T^{22} + 2542063 p^{30} T^{24} - 2106 p^{36} T^{26} + p^{42} T^{28} \) | |
| 13 | \( 1 - 17226 T^{2} + 11049203 p T^{4} - 786063335644 T^{6} + 3234975423221629 T^{8} - 830126042133420462 p T^{10} + \)\(30\!\cdots\!43\)\( T^{12} - \)\(72\!\cdots\!08\)\( T^{14} + \)\(30\!\cdots\!43\)\( p^{6} T^{16} - 830126042133420462 p^{13} T^{18} + 3234975423221629 p^{18} T^{20} - 786063335644 p^{24} T^{22} + 11049203 p^{31} T^{24} - 17226 p^{36} T^{26} + p^{42} T^{28} \) | |
| 17 | \( 1 - 37082 T^{2} + 715212171 T^{4} - 9393921400628 T^{6} + 93212173454486457 T^{8} - \)\(73\!\cdots\!58\)\( T^{10} + \)\(47\!\cdots\!95\)\( T^{12} - \)\(25\!\cdots\!72\)\( T^{14} + \)\(47\!\cdots\!95\)\( p^{6} T^{16} - \)\(73\!\cdots\!58\)\( p^{12} T^{18} + 93212173454486457 p^{18} T^{20} - 9393921400628 p^{24} T^{22} + 715212171 p^{30} T^{24} - 37082 p^{36} T^{26} + p^{42} T^{28} \) | |
| 19 | \( ( 1 + 26 T + 29977 T^{2} + 1444732 T^{3} + 410434113 T^{4} + 27322456630 T^{5} + 3648491456561 T^{6} + 255952746128840 T^{7} + 3648491456561 p^{3} T^{8} + 27322456630 p^{6} T^{9} + 410434113 p^{9} T^{10} + 1444732 p^{12} T^{11} + 29977 p^{15} T^{12} + 26 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 23 | \( 1 - 104054 T^{2} + 5018939647 T^{4} - 149242705388276 T^{6} + 3088159124512363021 T^{8} - \)\(48\!\cdots\!50\)\( T^{10} + \)\(63\!\cdots\!55\)\( T^{12} - \)\(77\!\cdots\!60\)\( T^{14} + \)\(63\!\cdots\!55\)\( p^{6} T^{16} - \)\(48\!\cdots\!50\)\( p^{12} T^{18} + 3088159124512363021 p^{18} T^{20} - 149242705388276 p^{24} T^{22} + 5018939647 p^{30} T^{24} - 104054 p^{36} T^{26} + p^{42} T^{28} \) | |
| 29 | \( ( 1 + 572 T + 273539 T^{2} + 87108760 T^{3} + 24251491677 T^{4} + 5374060095908 T^{5} + 1062807410319463 T^{6} + 175199251520296784 T^{7} + 1062807410319463 p^{3} T^{8} + 5374060095908 p^{6} T^{9} + 24251491677 p^{9} T^{10} + 87108760 p^{12} T^{11} + 273539 p^{15} T^{12} + 572 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 31 | \( ( 1 + 140 T + 99649 T^{2} + 11808136 T^{3} + 3845435853 T^{4} + 383291897332 T^{5} + 85357656204941 T^{6} + 9205444393625072 T^{7} + 85357656204941 p^{3} T^{8} + 383291897332 p^{6} T^{9} + 3845435853 p^{9} T^{10} + 11808136 p^{12} T^{11} + 99649 p^{15} T^{12} + 140 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 37 | \( 1 - 611150 T^{2} + 176053223715 T^{4} - 31777360586389292 T^{6} + \)\(40\!\cdots\!97\)\( T^{8} - \)\(38\!\cdots\!66\)\( T^{10} + \)\(27\!\cdots\!35\)\( T^{12} - \)\(15\!\cdots\!84\)\( T^{14} + \)\(27\!\cdots\!35\)\( p^{6} T^{16} - \)\(38\!\cdots\!66\)\( p^{12} T^{18} + \)\(40\!\cdots\!97\)\( p^{18} T^{20} - 31777360586389292 p^{24} T^{22} + 176053223715 p^{30} T^{24} - 611150 p^{36} T^{26} + p^{42} T^{28} \) | |
| 41 | \( ( 1 - 896 T + 596699 T^{2} - 283388128 T^{3} + 2825430201 p T^{4} - 39266584863872 T^{5} + 12150564909737299 T^{6} - 3308532939904287296 T^{7} + 12150564909737299 p^{3} T^{8} - 39266584863872 p^{6} T^{9} + 2825430201 p^{10} T^{10} - 283388128 p^{12} T^{11} + 596699 p^{15} T^{12} - 896 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 43 | \( 1 - 905982 T^{2} + 9004313401 p T^{4} - 104050185075892956 T^{6} + \)\(19\!\cdots\!37\)\( T^{8} - \)\(28\!\cdots\!34\)\( T^{10} + \)\(31\!\cdots\!03\)\( T^{12} - \)\(28\!\cdots\!36\)\( T^{14} + \)\(31\!\cdots\!03\)\( p^{6} T^{16} - \)\(28\!\cdots\!34\)\( p^{12} T^{18} + \)\(19\!\cdots\!37\)\( p^{18} T^{20} - 104050185075892956 p^{24} T^{22} + 9004313401 p^{31} T^{24} - 905982 p^{36} T^{26} + p^{42} T^{28} \) | |
| 47 | \( 1 - 1103126 T^{2} + 550480198479 T^{4} - 164398313886078068 T^{6} + \)\(32\!\cdots\!37\)\( T^{8} - \)\(48\!\cdots\!02\)\( T^{10} + \)\(56\!\cdots\!99\)\( T^{12} - \)\(59\!\cdots\!80\)\( T^{14} + \)\(56\!\cdots\!99\)\( p^{6} T^{16} - \)\(48\!\cdots\!02\)\( p^{12} T^{18} + \)\(32\!\cdots\!37\)\( p^{18} T^{20} - 164398313886078068 p^{24} T^{22} + 550480198479 p^{30} T^{24} - 1103126 p^{36} T^{26} + p^{42} T^{28} \) | |
| 53 | \( 1 - 1172898 T^{2} + 681558373223 T^{4} - 259947718209735868 T^{6} + \)\(73\!\cdots\!33\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{10} + \)\(30\!\cdots\!27\)\( T^{12} - \)\(48\!\cdots\!64\)\( T^{14} + \)\(30\!\cdots\!27\)\( p^{6} T^{16} - \)\(16\!\cdots\!02\)\( p^{12} T^{18} + \)\(73\!\cdots\!33\)\( p^{18} T^{20} - 259947718209735868 p^{24} T^{22} + 681558373223 p^{30} T^{24} - 1172898 p^{36} T^{26} + p^{42} T^{28} \) | |
| 59 | \( ( 1 + 1316 T + 1073101 T^{2} + 487955192 T^{3} + 194106073349 T^{4} + 80400028985020 T^{5} + 56221473691613793 T^{6} + 28197736893026094224 T^{7} + 56221473691613793 p^{3} T^{8} + 80400028985020 p^{6} T^{9} + 194106073349 p^{9} T^{10} + 487955192 p^{12} T^{11} + 1073101 p^{15} T^{12} + 1316 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 61 | \( ( 1 + 386 T + 974809 T^{2} + 282185992 T^{3} + 424153534803 T^{4} + 88205320620958 T^{5} + 120158895848851379 T^{6} + 19997641594322763344 T^{7} + 120158895848851379 p^{3} T^{8} + 88205320620958 p^{6} T^{9} + 424153534803 p^{9} T^{10} + 282185992 p^{12} T^{11} + 974809 p^{15} T^{12} + 386 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 67 | \( 1 - 31830 p T^{2} + 2466476183507 T^{4} - 1972773130799524660 T^{6} + \)\(12\!\cdots\!93\)\( T^{8} - \)\(58\!\cdots\!02\)\( T^{10} + \)\(23\!\cdots\!59\)\( T^{12} - \)\(76\!\cdots\!36\)\( T^{14} + \)\(23\!\cdots\!59\)\( p^{6} T^{16} - \)\(58\!\cdots\!02\)\( p^{12} T^{18} + \)\(12\!\cdots\!93\)\( p^{18} T^{20} - 1972773130799524660 p^{24} T^{22} + 2466476183507 p^{30} T^{24} - 31830 p^{37} T^{26} + p^{42} T^{28} \) | |
| 71 | \( ( 1 - 804 T + 1454947 T^{2} - 942577400 T^{3} + 968664238667 T^{4} - 510312219469948 T^{5} + 421027013945215113 T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + 421027013945215113 p^{3} T^{8} - 510312219469948 p^{6} T^{9} + 968664238667 p^{9} T^{10} - 942577400 p^{12} T^{11} + 1454947 p^{15} T^{12} - 804 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 73 | \( 1 - 2658782 T^{2} + 3881724191611 T^{4} - 3937234674807782940 T^{6} + \)\(30\!\cdots\!73\)\( T^{8} - \)\(18\!\cdots\!34\)\( T^{10} + \)\(96\!\cdots\!83\)\( T^{12} - \)\(41\!\cdots\!88\)\( T^{14} + \)\(96\!\cdots\!83\)\( p^{6} T^{16} - \)\(18\!\cdots\!34\)\( p^{12} T^{18} + \)\(30\!\cdots\!73\)\( p^{18} T^{20} - 3937234674807782940 p^{24} T^{22} + 3881724191611 p^{30} T^{24} - 2658782 p^{36} T^{26} + p^{42} T^{28} \) | |
| 79 | \( ( 1 - 374 T + 1756095 T^{2} - 649393768 T^{3} + 1642577778271 T^{4} - 568535463676106 T^{5} + 1095561494257081889 T^{6} - \)\(32\!\cdots\!56\)\( T^{7} + 1095561494257081889 p^{3} T^{8} - 568535463676106 p^{6} T^{9} + 1642577778271 p^{9} T^{10} - 649393768 p^{12} T^{11} + 1756095 p^{15} T^{12} - 374 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 83 | \( 1 - 4040642 T^{2} + 8234472722115 T^{4} - 11207776842765833156 T^{6} + \)\(11\!\cdots\!89\)\( T^{8} - \)\(94\!\cdots\!74\)\( T^{10} + \)\(65\!\cdots\!39\)\( T^{12} - \)\(39\!\cdots\!96\)\( T^{14} + \)\(65\!\cdots\!39\)\( p^{6} T^{16} - \)\(94\!\cdots\!74\)\( p^{12} T^{18} + \)\(11\!\cdots\!89\)\( p^{18} T^{20} - 11207776842765833156 p^{24} T^{22} + 8234472722115 p^{30} T^{24} - 4040642 p^{36} T^{26} + p^{42} T^{28} \) | |
| 89 | \( ( 1 + 694 T + 1888427 T^{2} + 1623972092 T^{3} + 2174197110417 T^{4} + 1642462505608906 T^{5} + 1884579985214566003 T^{6} + \)\(12\!\cdots\!04\)\( T^{7} + 1884579985214566003 p^{3} T^{8} + 1642462505608906 p^{6} T^{9} + 2174197110417 p^{9} T^{10} + 1623972092 p^{12} T^{11} + 1888427 p^{15} T^{12} + 694 p^{18} T^{13} + p^{21} T^{14} )^{2} \) | |
| 97 | \( 1 - 7273686 T^{2} + 26568099746059 T^{4} - 65304272575739116188 T^{6} + \)\(12\!\cdots\!93\)\( T^{8} - \)\(17\!\cdots\!66\)\( T^{10} + \)\(21\!\cdots\!27\)\( T^{12} - \)\(21\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!27\)\( p^{6} T^{16} - \)\(17\!\cdots\!66\)\( p^{12} T^{18} + \)\(12\!\cdots\!93\)\( p^{18} T^{20} - 65304272575739116188 p^{24} T^{22} + 26568099746059 p^{30} T^{24} - 7273686 p^{36} T^{26} + p^{42} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.54502537872088334310966291932, −3.37095339875559364468832979068, −3.29461736310518070961594161578, −3.27811647990656732106634727619, −2.85333523244002374614735114784, −2.84874681053484164987611479768, −2.80111716674834126557948707790, −2.67261792849289858284293578949, −2.65502933981127901800650998019, −2.46893575930620065398594452030, −2.41717118954405999329467564911, −2.18034405990190887935964657656, −2.09698344676926362900406906832, −1.92935693861648884834171464685, −1.92417794104894017203738357649, −1.87642891791202600340846309712, −1.86258163312498570321839552229, −1.50235188176998245923392535532, −1.15412435849413532610183426225, −0.74246766381527953038832249684, −0.72206694793231225683922860088, −0.45685257242214384611566328864, −0.44007397144152808874271054054, −0.37676581177375923113794615757, −0.02306275864800523768713098576, 0.02306275864800523768713098576, 0.37676581177375923113794615757, 0.44007397144152808874271054054, 0.45685257242214384611566328864, 0.72206694793231225683922860088, 0.74246766381527953038832249684, 1.15412435849413532610183426225, 1.50235188176998245923392535532, 1.86258163312498570321839552229, 1.87642891791202600340846309712, 1.92417794104894017203738357649, 1.92935693861648884834171464685, 2.09698344676926362900406906832, 2.18034405990190887935964657656, 2.41717118954405999329467564911, 2.46893575930620065398594452030, 2.65502933981127901800650998019, 2.67261792849289858284293578949, 2.80111716674834126557948707790, 2.84874681053484164987611479768, 2.85333523244002374614735114784, 3.27811647990656732106634727619, 3.29461736310518070961594161578, 3.37095339875559364468832979068, 3.54502537872088334310966291932
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.