Dirichlet series
| L(s) = 1 | + 1.41e7·3-s − 2.68e9·4-s − 1.45e12·7-s + 1.20e14·9-s − 3.79e16·12-s + 1.12e16·13-s + 4.32e18·16-s − 4.06e17·19-s − 2.06e19·21-s + 4.38e21·25-s + 5.96e20·27-s + 3.91e21·28-s − 3.85e21·31-s − 3.24e23·36-s + 6.91e23·37-s + 1.58e23·39-s − 1.05e25·43-s + 6.11e25·48-s − 1.18e26·49-s − 3.01e25·52-s − 5.74e24·57-s − 2.94e27·61-s − 1.76e26·63-s − 5.41e27·64-s − 1.05e28·67-s + 1.21e28·73-s + 6.20e28·75-s + ⋯ |
| L(s) = 1 | + 0.985·3-s − 5/2·4-s − 0.307·7-s + 0.586·9-s − 2.46·12-s + 0.219·13-s + 15/4·16-s − 0.0267·19-s − 0.302·21-s + 4.71·25-s + 0.201·27-s + 0.768·28-s − 0.164·31-s − 1.46·36-s + 2.07·37-s + 0.216·39-s − 3.33·43-s + 3.69·48-s − 5.27·49-s − 0.549·52-s − 0.0263·57-s − 4.89·61-s − 0.180·63-s − 4.37·64-s − 4.28·67-s + 1.35·73-s + 4.64·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 60466176 ^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60466176 ^{s/2} \, \Gamma_{\C}(s+15)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(60466176\) = \(2^{10} \cdot 3^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.19456\times 10^{15}\) |
| Root analytic conductor: | \(5.84880\) |
| Motivic weight: | \(30\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 60466176,\ (\ :[15]^{10}),\ 1)\) |
Particular Values
| \(L(\frac{31}{2})\) | \(\approx\) | \(0.02948237781\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02948237781\) |
| \(L(16)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{29} T^{2} )^{5} \) |
| 3 | \( 1 - 4713254 p T + 325555039399 p^{5} T^{2} - 3992288720392 p^{13} T^{3} + 268750343166296160 p^{23} T^{4} - 21542281297460613120 p^{35} T^{5} + 268750343166296160 p^{53} T^{6} - 3992288720392 p^{73} T^{7} + 325555039399 p^{95} T^{8} - 4713254 p^{121} T^{9} + p^{150} T^{10} \) | |
| good | 5 | \( 1 - \)\(87\!\cdots\!22\)\( p T^{2} + \)\(43\!\cdots\!61\)\( p^{2} T^{4} - \)\(94\!\cdots\!68\)\( p^{9} T^{6} + \)\(15\!\cdots\!86\)\( p^{16} T^{8} - \)\(16\!\cdots\!88\)\( p^{26} T^{10} + \)\(15\!\cdots\!86\)\( p^{76} T^{12} - \)\(94\!\cdots\!68\)\( p^{129} T^{14} + \)\(43\!\cdots\!61\)\( p^{182} T^{16} - \)\(87\!\cdots\!22\)\( p^{241} T^{18} + p^{300} T^{20} \) |
| 7 | \( ( 1 + 729667300598 T + \)\(85\!\cdots\!91\)\( p T^{2} - \)\(20\!\cdots\!92\)\( p^{3} T^{3} + \)\(14\!\cdots\!90\)\( p^{6} T^{4} - \)\(60\!\cdots\!16\)\( p^{8} T^{5} + \)\(14\!\cdots\!90\)\( p^{36} T^{6} - \)\(20\!\cdots\!92\)\( p^{63} T^{7} + \)\(85\!\cdots\!91\)\( p^{91} T^{8} + 729667300598 p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 11 | \( 1 - \)\(99\!\cdots\!50\)\( p T^{2} + \)\(59\!\cdots\!05\)\( T^{4} - \)\(17\!\cdots\!00\)\( p^{2} T^{6} + \)\(31\!\cdots\!10\)\( p^{6} T^{8} - \)\(43\!\cdots\!00\)\( p^{10} T^{10} + \)\(31\!\cdots\!10\)\( p^{66} T^{12} - \)\(17\!\cdots\!00\)\( p^{122} T^{14} + \)\(59\!\cdots\!05\)\( p^{180} T^{16} - \)\(99\!\cdots\!50\)\( p^{241} T^{18} + p^{300} T^{20} \) | |
| 13 | \( ( 1 - 5620928954228290 T + \)\(46\!\cdots\!65\)\( p T^{2} - \)\(87\!\cdots\!60\)\( p^{2} T^{3} + \)\(86\!\cdots\!10\)\( p^{4} T^{4} - \)\(26\!\cdots\!60\)\( p^{6} T^{5} + \)\(86\!\cdots\!10\)\( p^{34} T^{6} - \)\(87\!\cdots\!60\)\( p^{62} T^{7} + \)\(46\!\cdots\!65\)\( p^{91} T^{8} - 5620928954228290 p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 17 | \( 1 - \)\(27\!\cdots\!50\)\( T^{2} + \)\(33\!\cdots\!05\)\( T^{4} - \)\(11\!\cdots\!00\)\( p^{2} T^{6} + \)\(45\!\cdots\!10\)\( p^{4} T^{8} - \)\(15\!\cdots\!00\)\( p^{6} T^{10} + \)\(45\!\cdots\!10\)\( p^{64} T^{12} - \)\(11\!\cdots\!00\)\( p^{122} T^{14} + \)\(33\!\cdots\!05\)\( p^{180} T^{16} - \)\(27\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 19 | \( ( 1 + 203217470506219310 T + \)\(57\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!20\)\( p T^{3} + \)\(53\!\cdots\!10\)\( p^{2} T^{4} - \)\(19\!\cdots\!72\)\( p^{3} T^{5} + \)\(53\!\cdots\!10\)\( p^{32} T^{6} - \)\(11\!\cdots\!20\)\( p^{61} T^{7} + \)\(57\!\cdots\!45\)\( p^{90} T^{8} + 203217470506219310 p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 23 | \( 1 - \)\(35\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!45\)\( p^{2} T^{4} - \)\(25\!\cdots\!00\)\( p^{4} T^{6} + \)\(46\!\cdots\!90\)\( p^{6} T^{8} - \)\(69\!\cdots\!00\)\( p^{8} T^{10} + \)\(46\!\cdots\!90\)\( p^{66} T^{12} - \)\(25\!\cdots\!00\)\( p^{124} T^{14} + \)\(11\!\cdots\!45\)\( p^{182} T^{16} - \)\(35\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 29 | \( 1 - \)\(34\!\cdots\!10\)\( p^{2} T^{2} + \)\(75\!\cdots\!45\)\( p^{4} T^{4} - \)\(11\!\cdots\!20\)\( p^{6} T^{6} + \)\(14\!\cdots\!10\)\( p^{8} T^{8} - \)\(13\!\cdots\!52\)\( p^{10} T^{10} + \)\(14\!\cdots\!10\)\( p^{68} T^{12} - \)\(11\!\cdots\!20\)\( p^{126} T^{14} + \)\(75\!\cdots\!45\)\( p^{184} T^{16} - \)\(34\!\cdots\!10\)\( p^{242} T^{18} + p^{300} T^{20} \) | |
| 31 | \( ( 1 + \)\(19\!\cdots\!10\)\( T + \)\(19\!\cdots\!45\)\( T^{2} - \)\(53\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} - \)\(70\!\cdots\!48\)\( T^{5} + \)\(16\!\cdots\!10\)\( p^{30} T^{6} - \)\(53\!\cdots\!80\)\( p^{60} T^{7} + \)\(19\!\cdots\!45\)\( p^{90} T^{8} + \)\(19\!\cdots\!10\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 37 | \( ( 1 - \)\(34\!\cdots\!58\)\( T + \)\(39\!\cdots\!17\)\( T^{2} - \)\(43\!\cdots\!64\)\( T^{3} + \)\(91\!\cdots\!50\)\( T^{4} + \)\(27\!\cdots\!76\)\( T^{5} + \)\(91\!\cdots\!50\)\( p^{30} T^{6} - \)\(43\!\cdots\!64\)\( p^{60} T^{7} + \)\(39\!\cdots\!17\)\( p^{90} T^{8} - \)\(34\!\cdots\!58\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 41 | \( 1 - \)\(11\!\cdots\!50\)\( T^{2} + \)\(63\!\cdots\!05\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(70\!\cdots\!10\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{10} + \)\(70\!\cdots\!10\)\( p^{60} T^{12} - \)\(23\!\cdots\!00\)\( p^{120} T^{14} + \)\(63\!\cdots\!05\)\( p^{180} T^{16} - \)\(11\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 43 | \( ( 1 + \)\(52\!\cdots\!86\)\( T + \)\(26\!\cdots\!53\)\( T^{2} + \)\(10\!\cdots\!04\)\( T^{3} + \)\(36\!\cdots\!62\)\( T^{4} + \)\(96\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!62\)\( p^{30} T^{6} + \)\(10\!\cdots\!04\)\( p^{60} T^{7} + \)\(26\!\cdots\!53\)\( p^{90} T^{8} + \)\(52\!\cdots\!86\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 47 | \( 1 - \)\(92\!\cdots\!90\)\( T^{2} + \)\(41\!\cdots\!45\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{6} + \)\(26\!\cdots\!10\)\( T^{8} - \)\(44\!\cdots\!48\)\( T^{10} + \)\(26\!\cdots\!10\)\( p^{60} T^{12} - \)\(12\!\cdots\!80\)\( p^{120} T^{14} + \)\(41\!\cdots\!45\)\( p^{180} T^{16} - \)\(92\!\cdots\!90\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 53 | \( 1 - \)\(35\!\cdots\!50\)\( T^{2} + \)\(62\!\cdots\!05\)\( T^{4} - \)\(13\!\cdots\!00\)\( p T^{6} + \)\(58\!\cdots\!10\)\( T^{8} - \)\(36\!\cdots\!00\)\( T^{10} + \)\(58\!\cdots\!10\)\( p^{60} T^{12} - \)\(13\!\cdots\!00\)\( p^{121} T^{14} + \)\(62\!\cdots\!05\)\( p^{180} T^{16} - \)\(35\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 59 | \( 1 - \)\(62\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!05\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!10\)\( p^{60} T^{12} - \)\(54\!\cdots\!00\)\( p^{120} T^{14} + \)\(22\!\cdots\!05\)\( p^{180} T^{16} - \)\(62\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 61 | \( ( 1 + \)\(14\!\cdots\!30\)\( T + \)\(14\!\cdots\!65\)\( T^{2} + \)\(69\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} + \)\(22\!\cdots\!76\)\( T^{5} + \)\(22\!\cdots\!70\)\( p^{30} T^{6} + \)\(69\!\cdots\!80\)\( p^{60} T^{7} + \)\(14\!\cdots\!65\)\( p^{90} T^{8} + \)\(14\!\cdots\!30\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 67 | \( ( 1 + \)\(52\!\cdots\!22\)\( T + \)\(17\!\cdots\!77\)\( T^{2} + \)\(47\!\cdots\!36\)\( T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(49\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!10\)\( p^{30} T^{6} + \)\(47\!\cdots\!36\)\( p^{60} T^{7} + \)\(17\!\cdots\!77\)\( p^{90} T^{8} + \)\(52\!\cdots\!22\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 71 | \( 1 - \)\(24\!\cdots\!10\)\( T^{2} + \)\(27\!\cdots\!45\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(96\!\cdots\!10\)\( T^{8} - \)\(37\!\cdots\!52\)\( T^{10} + \)\(96\!\cdots\!10\)\( p^{60} T^{12} - \)\(19\!\cdots\!20\)\( p^{120} T^{14} + \)\(27\!\cdots\!45\)\( p^{180} T^{16} - \)\(24\!\cdots\!10\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 73 | \( ( 1 - \)\(60\!\cdots\!50\)\( T + \)\(25\!\cdots\!45\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!10\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!10\)\( p^{30} T^{6} - \)\(16\!\cdots\!00\)\( p^{60} T^{7} + \)\(25\!\cdots\!45\)\( p^{90} T^{8} - \)\(60\!\cdots\!50\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 79 | \( ( 1 - \)\(34\!\cdots\!10\)\( T + \)\(25\!\cdots\!45\)\( T^{2} - \)\(73\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!10\)\( T^{4} - \)\(75\!\cdots\!52\)\( T^{5} + \)\(33\!\cdots\!10\)\( p^{30} T^{6} - \)\(73\!\cdots\!20\)\( p^{60} T^{7} + \)\(25\!\cdots\!45\)\( p^{90} T^{8} - \)\(34\!\cdots\!10\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
| 83 | \( 1 - \)\(13\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!65\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{6} - \)\(31\!\cdots\!30\)\( T^{8} + \)\(25\!\cdots\!24\)\( T^{10} - \)\(31\!\cdots\!30\)\( p^{60} T^{12} - \)\(12\!\cdots\!80\)\( p^{120} T^{14} + \)\(67\!\cdots\!65\)\( p^{180} T^{16} - \)\(13\!\cdots\!30\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 89 | \( 1 - \)\(14\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!05\)\( T^{4} - \)\(62\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!10\)\( T^{8} - \)\(88\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!10\)\( p^{60} T^{12} - \)\(62\!\cdots\!00\)\( p^{120} T^{14} + \)\(11\!\cdots\!05\)\( p^{180} T^{16} - \)\(14\!\cdots\!50\)\( p^{240} T^{18} + p^{300} T^{20} \) | |
| 97 | \( ( 1 + \)\(69\!\cdots\!06\)\( T + \)\(15\!\cdots\!73\)\( T^{2} + \)\(85\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!62\)\( T^{4} + \)\(47\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!62\)\( p^{30} T^{6} + \)\(85\!\cdots\!84\)\( p^{60} T^{7} + \)\(15\!\cdots\!73\)\( p^{90} T^{8} + \)\(69\!\cdots\!06\)\( p^{120} T^{9} + p^{150} T^{10} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.40151143398295059417145939838, −4.38630975303757240960172814023, −3.92094097689586676541238045436, −3.76370482499257785293209984783, −3.42236521083788146418860987604, −3.36668454782786391946158674487, −3.29597723625859225392928034442, −3.21343794988076543233983882944, −3.14399946982935412902459771211, −2.86250269701745799081290968768, −2.55788912780043908623306979683, −2.52656601871406933873072569557, −2.42091599862527068717879399475, −1.97181369551006165200223903492, −1.65263476144042656786008192064, −1.57976294701497699804968742945, −1.41386675328181599449966430386, −1.35472184486527097452845827072, −1.15989718197870355791219246871, −1.05881147999772516208583087233, −0.802346225829123791320482572508, −0.69974980213164739090286033849, −0.20932013313271428401353253124, −0.15055499545659577903887915759, −0.03624146001042523373188217199, 0.03624146001042523373188217199, 0.15055499545659577903887915759, 0.20932013313271428401353253124, 0.69974980213164739090286033849, 0.802346225829123791320482572508, 1.05881147999772516208583087233, 1.15989718197870355791219246871, 1.35472184486527097452845827072, 1.41386675328181599449966430386, 1.57976294701497699804968742945, 1.65263476144042656786008192064, 1.97181369551006165200223903492, 2.42091599862527068717879399475, 2.52656601871406933873072569557, 2.55788912780043908623306979683, 2.86250269701745799081290968768, 3.14399946982935412902459771211, 3.21343794988076543233983882944, 3.29597723625859225392928034442, 3.36668454782786391946158674487, 3.42236521083788146418860987604, 3.76370482499257785293209984783, 3.92094097689586676541238045436, 4.38630975303757240960172814023, 4.40151143398295059417145939838
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.