Dirichlet series
| L(s) = 1 | + 2.43e3·3-s + 1.03e4·4-s + 2.64e4·5-s + 6.87e5·9-s − 1.71e6·11-s + 2.51e7·12-s + 6.45e7·15-s + 3.67e7·16-s + 2.73e8·20-s + 3.30e8·23-s − 1.08e9·25-s − 3.01e9·27-s − 1.92e9·31-s − 4.18e9·33-s + 7.09e9·36-s + 1.78e9·37-s − 1.76e10·44-s + 1.82e10·45-s − 2.49e10·47-s + 8.95e10·48-s + 6.60e10·49-s − 1.63e10·53-s − 4.54e10·55-s + 6.23e10·59-s + 6.65e11·60-s + 4.21e10·64-s − 9.00e10·67-s + ⋯ |
| L(s) = 1 | + 3.34·3-s + 2.51·4-s + 1.69·5-s + 1.29·9-s − 0.968·11-s + 8.40·12-s + 5.66·15-s + 2.19·16-s + 4.26·20-s + 2.23·23-s − 4.46·25-s − 7.79·27-s − 2.16·31-s − 3.23·33-s + 3.25·36-s + 0.697·37-s − 2.43·44-s + 2.19·45-s − 2.31·47-s + 7.32·48-s + 4.77·49-s − 0.736·53-s − 1.64·55-s + 1.47·59-s + 14.2·60-s + 0.613·64-s − 0.995·67-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{10}\right)^{s/2} \, \Gamma_{\C}(s+6)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(11^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.05525\times 10^{10}\) |
| Root analytic conductor: | \(3.17079\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 11^{10} ,\ ( \ : [6]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(10.91815922\) |
| \(L(\frac12)\) | \(\approx\) | \(10.91815922\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( 1 + 156034 p T - 3752383617 p^{3} T^{2} - 9929148802328 p^{6} T^{3} + 4287314380300486 p^{9} T^{4} + 2114113673027284480 p^{13} T^{5} + 4287314380300486 p^{21} T^{6} - 9929148802328 p^{30} T^{7} - 3752383617 p^{39} T^{8} + 156034 p^{49} T^{9} + p^{60} T^{10} \) |
| good | 2 | \( 1 - 5153 p T^{2} + 2170299 p^{5} T^{4} - 2960968911 p^{7} T^{6} + 1009436172207 p^{11} T^{8} - 35513684952303 p^{18} T^{10} + 1009436172207 p^{35} T^{12} - 2960968911 p^{55} T^{14} + 2170299 p^{77} T^{16} - 5153 p^{97} T^{18} + p^{120} T^{20} \) |
| 3 | \( ( 1 - 406 p T + 627097 p T^{2} - 7205620 p^{5} T^{3} + 2436567997 p^{6} T^{4} - 65716296938 p^{9} T^{5} + 2436567997 p^{18} T^{6} - 7205620 p^{29} T^{7} + 627097 p^{37} T^{8} - 406 p^{49} T^{9} + p^{60} T^{10} )^{2} \) | |
| 5 | \( ( 1 - 13246 T + 32307963 p^{2} T^{2} - 202264294064 p^{2} T^{3} + 418008025891661 p^{4} T^{4} - 56172293545340166 p^{6} T^{5} + 418008025891661 p^{16} T^{6} - 202264294064 p^{26} T^{7} + 32307963 p^{38} T^{8} - 13246 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 7 | \( 1 - 66043580506 T^{2} + \)\(21\!\cdots\!33\)\( T^{4} - \)\(68\!\cdots\!04\)\( p T^{6} + \)\(23\!\cdots\!02\)\( p^{3} T^{8} - \)\(71\!\cdots\!96\)\( p^{5} T^{10} + \)\(23\!\cdots\!02\)\( p^{27} T^{12} - \)\(68\!\cdots\!04\)\( p^{49} T^{14} + \)\(21\!\cdots\!33\)\( p^{72} T^{16} - 66043580506 p^{96} T^{18} + p^{120} T^{20} \) | |
| 13 | \( 1 - 63177329499226 T^{2} + \)\(19\!\cdots\!13\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} + \)\(48\!\cdots\!68\)\( T^{10} + \)\(12\!\cdots\!86\)\( p^{24} T^{12} - \)\(33\!\cdots\!08\)\( p^{48} T^{14} + \)\(19\!\cdots\!13\)\( p^{72} T^{16} - 63177329499226 p^{96} T^{18} + p^{120} T^{20} \) | |
| 17 | \( 1 - 1526678011493146 T^{2} + \)\(11\!\cdots\!33\)\( T^{4} - \)\(58\!\cdots\!68\)\( T^{6} + \)\(17\!\cdots\!46\)\( T^{8} - \)\(52\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!46\)\( p^{24} T^{12} - \)\(58\!\cdots\!68\)\( p^{48} T^{14} + \)\(11\!\cdots\!33\)\( p^{72} T^{16} - 1526678011493146 p^{96} T^{18} + p^{120} T^{20} \) | |
| 19 | \( 1 - 5117141354901850 T^{2} + \)\(20\!\cdots\!25\)\( T^{4} - \)\(75\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!50\)\( T^{8} - \)\(49\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!50\)\( p^{24} T^{12} - \)\(75\!\cdots\!00\)\( p^{48} T^{14} + \)\(20\!\cdots\!25\)\( p^{72} T^{16} - 5117141354901850 p^{96} T^{18} + p^{120} T^{20} \) | |
| 23 | \( ( 1 - 165148738 T + 4351770058698237 p T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!33\)\( T^{4} - \)\(41\!\cdots\!34\)\( T^{5} + \)\(41\!\cdots\!33\)\( p^{12} T^{6} - \)\(12\!\cdots\!00\)\( p^{24} T^{7} + 4351770058698237 p^{37} T^{8} - 165148738 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 29 | \( 1 - 1145013891598603210 T^{2} + \)\(94\!\cdots\!05\)\( T^{4} - \)\(57\!\cdots\!80\)\( T^{6} + \)\(27\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(27\!\cdots\!10\)\( p^{24} T^{12} - \)\(57\!\cdots\!80\)\( p^{48} T^{14} + \)\(94\!\cdots\!05\)\( p^{72} T^{16} - 1145013891598603210 p^{96} T^{18} + p^{120} T^{20} \) | |
| 31 | \( ( 1 + 960977774 T + 2301502490439379563 T^{2} + \)\(25\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} + \)\(27\!\cdots\!38\)\( T^{5} + \)\(30\!\cdots\!81\)\( p^{12} T^{6} + \)\(25\!\cdots\!12\)\( p^{24} T^{7} + 2301502490439379563 p^{36} T^{8} + 960977774 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 37 | \( ( 1 - 894161998 T + 25790259709692783411 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!93\)\( T^{4} - \)\(56\!\cdots\!54\)\( T^{5} + \)\(28\!\cdots\!93\)\( p^{12} T^{6} - \)\(10\!\cdots\!80\)\( p^{24} T^{7} + 25790259709692783411 p^{36} T^{8} - 894161998 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 41 | \( 1 - \)\(15\!\cdots\!10\)\( T^{2} + \)\(11\!\cdots\!05\)\( T^{4} - \)\(58\!\cdots\!80\)\( T^{6} + \)\(20\!\cdots\!10\)\( T^{8} - \)\(53\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( p^{24} T^{12} - \)\(58\!\cdots\!80\)\( p^{48} T^{14} + \)\(11\!\cdots\!05\)\( p^{72} T^{16} - \)\(15\!\cdots\!10\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 43 | \( 1 - \)\(29\!\cdots\!06\)\( T^{2} + \)\(41\!\cdots\!33\)\( T^{4} - \)\(37\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(10\!\cdots\!72\)\( T^{10} + \)\(23\!\cdots\!86\)\( p^{24} T^{12} - \)\(37\!\cdots\!28\)\( p^{48} T^{14} + \)\(41\!\cdots\!33\)\( p^{72} T^{16} - \)\(29\!\cdots\!06\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 47 | \( ( 1 + 12487755062 T + \)\(28\!\cdots\!81\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!58\)\( p^{12} T^{6} + \)\(10\!\cdots\!40\)\( p^{24} T^{7} + \)\(28\!\cdots\!81\)\( p^{36} T^{8} + 12487755062 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 53 | \( ( 1 + 8162751062 T + \)\(20\!\cdots\!41\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!38\)\( T^{4} + \)\(96\!\cdots\!36\)\( T^{5} + \)\(18\!\cdots\!38\)\( p^{12} T^{6} + \)\(13\!\cdots\!40\)\( p^{24} T^{7} + \)\(20\!\cdots\!41\)\( p^{36} T^{8} + 8162751062 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 59 | \( ( 1 - 31169695282 T + \)\(73\!\cdots\!47\)\( T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!57\)\( T^{4} - \)\(47\!\cdots\!02\)\( T^{5} + \)\(24\!\cdots\!57\)\( p^{12} T^{6} - \)\(18\!\cdots\!12\)\( p^{24} T^{7} + \)\(73\!\cdots\!47\)\( p^{36} T^{8} - 31169695282 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 61 | \( 1 - \)\(19\!\cdots\!50\)\( T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(41\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{10} + \)\(41\!\cdots\!50\)\( p^{24} T^{12} - \)\(10\!\cdots\!00\)\( p^{48} T^{14} + \)\(17\!\cdots\!25\)\( p^{72} T^{16} - \)\(19\!\cdots\!50\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 67 | \( ( 1 + 45017650622 T + \)\(32\!\cdots\!11\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(46\!\cdots\!33\)\( T^{4} + \)\(15\!\cdots\!26\)\( T^{5} + \)\(46\!\cdots\!33\)\( p^{12} T^{6} + \)\(13\!\cdots\!00\)\( p^{24} T^{7} + \)\(32\!\cdots\!11\)\( p^{36} T^{8} + 45017650622 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 179955059870 T + \)\(56\!\cdots\!95\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!05\)\( T^{4} + \)\(14\!\cdots\!98\)\( T^{5} + \)\(14\!\cdots\!05\)\( p^{12} T^{6} + \)\(71\!\cdots\!80\)\( p^{24} T^{7} + \)\(56\!\cdots\!95\)\( p^{36} T^{8} + 179955059870 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 73 | \( 1 - \)\(98\!\cdots\!26\)\( T^{2} + \)\(51\!\cdots\!13\)\( T^{4} - \)\(19\!\cdots\!28\)\( T^{6} + \)\(60\!\cdots\!06\)\( T^{8} - \)\(15\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!06\)\( p^{24} T^{12} - \)\(19\!\cdots\!28\)\( p^{48} T^{14} + \)\(51\!\cdots\!13\)\( p^{72} T^{16} - \)\(98\!\cdots\!26\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 79 | \( 1 - \)\(38\!\cdots\!70\)\( T^{2} + \)\(73\!\cdots\!85\)\( T^{4} - \)\(92\!\cdots\!60\)\( T^{6} + \)\(83\!\cdots\!70\)\( T^{8} - \)\(56\!\cdots\!52\)\( T^{10} + \)\(83\!\cdots\!70\)\( p^{24} T^{12} - \)\(92\!\cdots\!60\)\( p^{48} T^{14} + \)\(73\!\cdots\!85\)\( p^{72} T^{16} - \)\(38\!\cdots\!70\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 83 | \( 1 - \)\(40\!\cdots\!26\)\( T^{2} + \)\(96\!\cdots\!53\)\( T^{4} - \)\(16\!\cdots\!88\)\( T^{6} + \)\(22\!\cdots\!26\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{10} + \)\(22\!\cdots\!26\)\( p^{24} T^{12} - \)\(16\!\cdots\!88\)\( p^{48} T^{14} + \)\(96\!\cdots\!53\)\( p^{72} T^{16} - \)\(40\!\cdots\!26\)\( p^{96} T^{18} + p^{120} T^{20} \) | |
| 89 | \( ( 1 - 335498390206 T + \)\(10\!\cdots\!03\)\( T^{2} - \)\(27\!\cdots\!68\)\( T^{3} + \)\(44\!\cdots\!01\)\( T^{4} - \)\(94\!\cdots\!62\)\( T^{5} + \)\(44\!\cdots\!01\)\( p^{12} T^{6} - \)\(27\!\cdots\!68\)\( p^{24} T^{7} + \)\(10\!\cdots\!03\)\( p^{36} T^{8} - 335498390206 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| 97 | \( ( 1 + 3125352342482 T + \)\(61\!\cdots\!91\)\( T^{2} + \)\(86\!\cdots\!40\)\( T^{3} + \)\(96\!\cdots\!13\)\( T^{4} + \)\(87\!\cdots\!46\)\( T^{5} + \)\(96\!\cdots\!13\)\( p^{12} T^{6} + \)\(86\!\cdots\!40\)\( p^{24} T^{7} + \)\(61\!\cdots\!91\)\( p^{36} T^{8} + 3125352342482 p^{48} T^{9} + p^{60} T^{10} )^{2} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.06503730892599228626762514641, −5.73973409413146236980687405803, −5.62173049209927548083272526846, −5.46241796609488694677591418762, −5.34489695762179773341783113572, −5.25060057668144635018352723148, −4.65362692343832210666688688755, −4.06449061974704228475607615387, −4.06253083983408515216903363972, −3.75394629712621173613568732013, −3.63428885272791745977302568964, −3.08004267295960256304613056352, −3.05489240101678304621905459590, −2.67005105637449006321236845487, −2.66094414196427942264989929428, −2.61772200418506616878449664383, −2.61142004654666547042672433941, −2.14189057731434925897486088553, −1.98853782769296521458150469022, −1.73080062762996615624344506108, −1.61333613093698540445133924769, −1.44475219909566298410972226948, −0.74077947015724377286071077394, −0.28129380724543025112098130813, −0.19533154911075082340345238065, 0.19533154911075082340345238065, 0.28129380724543025112098130813, 0.74077947015724377286071077394, 1.44475219909566298410972226948, 1.61333613093698540445133924769, 1.73080062762996615624344506108, 1.98853782769296521458150469022, 2.14189057731434925897486088553, 2.61142004654666547042672433941, 2.61772200418506616878449664383, 2.66094414196427942264989929428, 2.67005105637449006321236845487, 3.05489240101678304621905459590, 3.08004267295960256304613056352, 3.63428885272791745977302568964, 3.75394629712621173613568732013, 4.06253083983408515216903363972, 4.06449061974704228475607615387, 4.65362692343832210666688688755, 5.25060057668144635018352723148, 5.34489695762179773341783113572, 5.46241796609488694677591418762, 5.62173049209927548083272526846, 5.73973409413146236980687405803, 6.06503730892599228626762514641
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.