Dirichlet series
| L(s) = 1 | + 22·2-s − 80·4-s + 3.11e3·5-s + 1.41e3·8-s − 9.84e4·9-s + 6.85e4·10-s − 2.85e5·13-s + 1.02e6·16-s + 2.26e6·17-s − 2.16e6·18-s − 2.49e5·20-s − 3.24e7·25-s − 6.27e6·26-s − 4.66e7·29-s + 3.07e6·32-s + 4.98e7·34-s + 7.87e6·36-s − 1.94e8·37-s + 4.41e6·40-s + 2.76e8·41-s − 3.06e8·45-s + 1.73e9·49-s − 7.13e8·50-s + 2.28e7·52-s − 1.52e9·53-s − 1.02e9·58-s − 3.48e9·61-s + ⋯ |
| L(s) = 1 | + 0.687·2-s − 0.0781·4-s + 0.997·5-s + 0.0432·8-s − 5/3·9-s + 0.685·10-s − 0.767·13-s + 0.981·16-s + 1.59·17-s − 1.14·18-s − 0.0778·20-s − 3.32·25-s − 0.527·26-s − 2.27·29-s + 0.0916·32-s + 1.09·34-s + 0.130·36-s − 2.80·37-s + 0.0430·40-s + 2.38·41-s − 1.66·45-s + 6.15·49-s − 2.28·50-s + 0.0599·52-s − 3.64·53-s − 1.56·58-s − 4.12·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10}\right)^{s/2} \, \Gamma_{\C}(s+5)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.63731\times 10^{8}\) |
| Root analytic conductor: | \(2.76121\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{10} ,\ ( \ : [5]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(0.3647480581\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3647480581\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - 11 p T + 141 p^{2} T^{2} - 487 p^{5} T^{3} - 2383 p^{8} T^{4} + 3777 p^{13} T^{5} - 2383 p^{18} T^{6} - 487 p^{25} T^{7} + 141 p^{32} T^{8} - 11 p^{41} T^{9} + p^{50} T^{10} \) |
| 3 | \( ( 1 + p^{9} T^{2} )^{5} \) | |
| good | 5 | \( ( 1 - 1558 T + 3973377 p T^{2} - 4805248936 p T^{3} + 8544452854466 p^{2} T^{4} - 9669638754857508 p^{2} T^{5} + 8544452854466 p^{12} T^{6} - 4805248936 p^{21} T^{7} + 3973377 p^{31} T^{8} - 1558 p^{40} T^{9} + p^{50} T^{10} )^{2} \) |
| 7 | \( 1 - 1739188858 T^{2} + 1516953460445315085 T^{4} - \)\(17\!\cdots\!92\)\( p^{2} T^{6} + \)\(15\!\cdots\!94\)\( p^{4} T^{8} - \)\(10\!\cdots\!80\)\( p^{6} T^{10} + \)\(15\!\cdots\!94\)\( p^{24} T^{12} - \)\(17\!\cdots\!92\)\( p^{42} T^{14} + 1516953460445315085 p^{60} T^{16} - 1739188858 p^{80} T^{18} + p^{100} T^{20} \) | |
| 11 | \( 1 - 81809175946 T^{2} + \)\(39\!\cdots\!37\)\( T^{4} - \)\(16\!\cdots\!48\)\( T^{6} + \)\(53\!\cdots\!14\)\( T^{8} - \)\(14\!\cdots\!08\)\( T^{10} + \)\(53\!\cdots\!14\)\( p^{20} T^{12} - \)\(16\!\cdots\!48\)\( p^{40} T^{14} + \)\(39\!\cdots\!37\)\( p^{60} T^{16} - 81809175946 p^{80} T^{18} + p^{100} T^{20} \) | |
| 13 | \( ( 1 + 142550 T + 333741070869 T^{2} + 23921333114807496 T^{3} + \)\(65\!\cdots\!30\)\( T^{4} + \)\(42\!\cdots\!96\)\( T^{5} + \)\(65\!\cdots\!30\)\( p^{10} T^{6} + 23921333114807496 p^{20} T^{7} + 333741070869 p^{30} T^{8} + 142550 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 17 | \( ( 1 - 1132210 T + 4116414384141 T^{2} - 2117920716241447640 T^{3} + \)\(47\!\cdots\!86\)\( T^{4} - \)\(77\!\cdots\!80\)\( T^{5} + \)\(47\!\cdots\!86\)\( p^{10} T^{6} - 2117920716241447640 p^{20} T^{7} + 4116414384141 p^{30} T^{8} - 1132210 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 19 | \( 1 - 29424277025002 T^{2} + \)\(37\!\cdots\!85\)\( T^{4} - \)\(26\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!94\)\( T^{8} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!94\)\( p^{20} T^{12} - \)\(26\!\cdots\!12\)\( p^{40} T^{14} + \)\(37\!\cdots\!85\)\( p^{60} T^{16} - 29424277025002 p^{80} T^{18} + p^{100} T^{20} \) | |
| 23 | \( 1 - 118697788171306 T^{2} + \)\(82\!\cdots\!37\)\( T^{4} - \)\(42\!\cdots\!68\)\( T^{6} + \)\(20\!\cdots\!62\)\( T^{8} - \)\(89\!\cdots\!24\)\( T^{10} + \)\(20\!\cdots\!62\)\( p^{20} T^{12} - \)\(42\!\cdots\!68\)\( p^{40} T^{14} + \)\(82\!\cdots\!37\)\( p^{60} T^{16} - 118697788171306 p^{80} T^{18} + p^{100} T^{20} \) | |
| 29 | \( ( 1 + 23336354 T + 1163414307352821 T^{2} + \)\(19\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!62\)\( T^{4} + \)\(84\!\cdots\!92\)\( T^{5} + \)\(51\!\cdots\!62\)\( p^{10} T^{6} + \)\(19\!\cdots\!72\)\( p^{20} T^{7} + 1163414307352821 p^{30} T^{8} + 23336354 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 31 | \( 1 - 4714422803758234 T^{2} + \)\(11\!\cdots\!89\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(22\!\cdots\!86\)\( T^{8} - \)\(20\!\cdots\!12\)\( T^{10} + \)\(22\!\cdots\!86\)\( p^{20} T^{12} - \)\(18\!\cdots\!12\)\( p^{40} T^{14} + \)\(11\!\cdots\!89\)\( p^{60} T^{16} - 4714422803758234 p^{80} T^{18} + p^{100} T^{20} \) | |
| 37 | \( ( 1 + 97233950 T + 18415614078065541 T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} + \)\(74\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!34\)\( p^{10} T^{6} + \)\(12\!\cdots\!12\)\( p^{20} T^{7} + 18415614078065541 p^{30} T^{8} + 97233950 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 41 | \( ( 1 - 138113890 T + 889807758147381 p T^{2} - \)\(17\!\cdots\!84\)\( T^{3} + \)\(34\!\cdots\!34\)\( T^{4} + \)\(53\!\cdots\!80\)\( T^{5} + \)\(34\!\cdots\!34\)\( p^{10} T^{6} - \)\(17\!\cdots\!84\)\( p^{20} T^{7} + 889807758147381 p^{31} T^{8} - 138113890 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 43 | \( 1 - 80925657641789962 T^{2} + \)\(30\!\cdots\!93\)\( T^{4} - \)\(85\!\cdots\!04\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{8} - \)\(29\!\cdots\!80\)\( p^{2} T^{10} + \)\(22\!\cdots\!94\)\( p^{20} T^{12} - \)\(85\!\cdots\!04\)\( p^{40} T^{14} + \)\(30\!\cdots\!93\)\( p^{60} T^{16} - 80925657641789962 p^{80} T^{18} + p^{100} T^{20} \) | |
| 47 | \( 1 - 345714433059785098 T^{2} + \)\(58\!\cdots\!49\)\( T^{4} - \)\(13\!\cdots\!12\)\( p T^{6} + \)\(51\!\cdots\!86\)\( T^{8} - \)\(30\!\cdots\!04\)\( T^{10} + \)\(51\!\cdots\!86\)\( p^{20} T^{12} - \)\(13\!\cdots\!12\)\( p^{41} T^{14} + \)\(58\!\cdots\!49\)\( p^{60} T^{16} - 345714433059785098 p^{80} T^{18} + p^{100} T^{20} \) | |
| 53 | \( ( 1 + 761360978 T + 902692782283934373 T^{2} + \)\(53\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!26\)\( T^{4} + \)\(13\!\cdots\!84\)\( T^{5} + \)\(32\!\cdots\!26\)\( p^{10} T^{6} + \)\(53\!\cdots\!12\)\( p^{20} T^{7} + 902692782283934373 p^{30} T^{8} + 761360978 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 59 | \( 1 - 3435272300044464202 T^{2} + \)\(56\!\cdots\!05\)\( T^{4} - \)\(59\!\cdots\!12\)\( T^{6} + \)\(44\!\cdots\!74\)\( T^{8} - \)\(25\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!74\)\( p^{20} T^{12} - \)\(59\!\cdots\!12\)\( p^{40} T^{14} + \)\(56\!\cdots\!05\)\( p^{60} T^{16} - 3435272300044464202 p^{80} T^{18} + p^{100} T^{20} \) | |
| 61 | \( ( 1 + 1744062302 T + 2120268786229627701 T^{2} + \)\(19\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!82\)\( T^{4} + \)\(16\!\cdots\!72\)\( T^{5} + \)\(19\!\cdots\!82\)\( p^{10} T^{6} + \)\(19\!\cdots\!96\)\( p^{20} T^{7} + 2120268786229627701 p^{30} T^{8} + 1744062302 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 67 | \( 1 - 7176172579914925738 T^{2} + \)\(31\!\cdots\!25\)\( T^{4} - \)\(15\!\cdots\!84\)\( p T^{6} + \)\(26\!\cdots\!94\)\( T^{8} - \)\(52\!\cdots\!40\)\( T^{10} + \)\(26\!\cdots\!94\)\( p^{20} T^{12} - \)\(15\!\cdots\!84\)\( p^{41} T^{14} + \)\(31\!\cdots\!25\)\( p^{60} T^{16} - 7176172579914925738 p^{80} T^{18} + p^{100} T^{20} \) | |
| 71 | \( 1 - 19470831043567567978 T^{2} + \)\(19\!\cdots\!97\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(62\!\cdots\!46\)\( T^{8} - \)\(23\!\cdots\!52\)\( T^{10} + \)\(62\!\cdots\!46\)\( p^{20} T^{12} - \)\(12\!\cdots\!40\)\( p^{40} T^{14} + \)\(19\!\cdots\!97\)\( p^{60} T^{16} - 19470831043567567978 p^{80} T^{18} + p^{100} T^{20} \) | |
| 73 | \( ( 1 - 1545259354 T + 14813285739003117309 T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!34\)\( p T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!34\)\( p^{11} T^{6} - \)\(18\!\cdots\!80\)\( p^{20} T^{7} + 14813285739003117309 p^{30} T^{8} - 1545259354 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 79 | \( 1 - 43626985337445416410 T^{2} + \)\(10\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(23\!\cdots\!86\)\( p^{20} T^{12} - \)\(18\!\cdots\!20\)\( p^{40} T^{14} + \)\(10\!\cdots\!49\)\( p^{60} T^{16} - 43626985337445416410 p^{80} T^{18} + p^{100} T^{20} \) | |
| 83 | \( 1 - \)\(10\!\cdots\!50\)\( T^{2} + \)\(52\!\cdots\!69\)\( T^{4} - \)\(15\!\cdots\!32\)\( T^{6} + \)\(31\!\cdots\!78\)\( T^{8} - \)\(53\!\cdots\!88\)\( T^{10} + \)\(31\!\cdots\!78\)\( p^{20} T^{12} - \)\(15\!\cdots\!32\)\( p^{40} T^{14} + \)\(52\!\cdots\!69\)\( p^{60} T^{16} - \)\(10\!\cdots\!50\)\( p^{80} T^{18} + p^{100} T^{20} \) | |
| 89 | \( ( 1 + 6835032998 T + 89879367559527150621 T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!82\)\( T^{4} + \)\(20\!\cdots\!16\)\( T^{5} + \)\(40\!\cdots\!82\)\( p^{10} T^{6} + \)\(53\!\cdots\!00\)\( p^{20} T^{7} + 89879367559527150621 p^{30} T^{8} + 6835032998 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 97 | \( ( 1 + 9402538742 T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(20\!\cdots\!24\)\( T^{3} + \)\(36\!\cdots\!94\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(36\!\cdots\!94\)\( p^{10} T^{6} + \)\(20\!\cdots\!24\)\( p^{20} T^{7} + \)\(28\!\cdots\!17\)\( p^{30} T^{8} + 9402538742 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 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
−6.21150214137311266403283641072, −5.96532376756603396230855044684, −5.88692064970671641929065858983, −5.66831885182708027051237255811, −5.46592735916751138691697651598, −5.45799866919511492209356164656, −5.36029513724611677747456468853, −4.74235980343347351160308413630, −4.63536039771453169674685413973, −4.43317632919388651385977558447, −4.00437673161602418829552720688, −3.76947756974037600852428516484, −3.52211358246456617170723073223, −3.44246072556503534055589385562, −3.27693799340222866447681489334, −2.68296598494653475776363379211, −2.59310329783223099316908891974, −2.19554952260181176885782899248, −2.07476211303843169152461290136, −1.73060297302932469729551524657, −1.46377424660197815134501443289, −1.24824244686522097284930966811, −0.67601849070590422942463913406, −0.41287979221803733959971612283, −0.06374073562910415071325194839, 0.06374073562910415071325194839, 0.41287979221803733959971612283, 0.67601849070590422942463913406, 1.24824244686522097284930966811, 1.46377424660197815134501443289, 1.73060297302932469729551524657, 2.07476211303843169152461290136, 2.19554952260181176885782899248, 2.59310329783223099316908891974, 2.68296598494653475776363379211, 3.27693799340222866447681489334, 3.44246072556503534055589385562, 3.52211358246456617170723073223, 3.76947756974037600852428516484, 4.00437673161602418829552720688, 4.43317632919388651385977558447, 4.63536039771453169674685413973, 4.74235980343347351160308413630, 5.36029513724611677747456468853, 5.45799866919511492209356164656, 5.46592735916751138691697651598, 5.66831885182708027051237255811, 5.88692064970671641929065858983, 5.96532376756603396230855044684, 6.21150214137311266403283641072
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.