Dirichlet series
| L(s) = 1 | − 2.13e7·3-s + 8.57e9·4-s − 5.56e12·7-s − 1.66e14·9-s − 1.83e17·12-s + 5.67e17·13-s + 3.24e19·16-s + 5.71e20·19-s + 1.19e20·21-s + 8.19e22·25-s − 1.75e22·27-s − 4.77e22·28-s + 2.33e24·31-s − 1.42e24·36-s − 3.53e25·37-s − 1.21e25·39-s + 2.01e25·43-s − 6.94e26·48-s − 6.89e27·49-s + 4.87e27·52-s − 1.22e28·57-s − 1.96e29·61-s + 9.26e26·63-s + 1.18e28·64-s − 4.42e29·67-s − 4.53e30·73-s − 1.75e30·75-s + ⋯ |
| L(s) = 1 | − 0.496·3-s + 1.99·4-s − 0.167·7-s − 0.0897·9-s − 0.992·12-s + 0.853·13-s + 1.76·16-s + 1.97·19-s + 0.0832·21-s + 3.52·25-s − 0.219·27-s − 0.334·28-s + 3.21·31-s − 0.179·36-s − 2.86·37-s − 0.423·39-s + 0.147·43-s − 0.874·48-s − 6.23·49-s + 1.70·52-s − 0.983·57-s − 5.35·61-s + 0.0150·63-s + 0.149·64-s − 2.68·67-s − 6.96·73-s − 1.74·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+16)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(59049\) = \(3^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.78802\times 10^{12}\) |
| Root analytic conductor: | \(4.41134\) |
| Motivic weight: | \(32\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 59049,\ (\ :[16]^{10}),\ 1)\) |
Particular Values
| \(L(\frac{33}{2})\) | \(\approx\) | \(4.255100013\) |
| \(L(\frac12)\) | \(\approx\) | \(4.255100013\) |
| \(L(17)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 2376350 p^{2} T + 285215772583 p^{7} T^{2} + 799186033576360 p^{16} T^{3} + 106347409409879888 p^{27} T^{4} + 14166463123098298368 p^{40} T^{5} + 106347409409879888 p^{59} T^{6} + 799186033576360 p^{80} T^{7} + 285215772583 p^{103} T^{8} + 2376350 p^{130} T^{9} + p^{160} T^{10} \) |
| good | 2 | \( 1 - 1072352381 p^{3} T^{2} + 40164943604147613 p^{10} T^{4} - \)\(64\!\cdots\!61\)\( p^{27} T^{6} + \)\(62\!\cdots\!13\)\( p^{42} T^{8} - \)\(25\!\cdots\!01\)\( p^{55} T^{10} + \)\(62\!\cdots\!13\)\( p^{106} T^{12} - \)\(64\!\cdots\!61\)\( p^{155} T^{14} + 40164943604147613 p^{202} T^{16} - 1072352381 p^{259} T^{18} + p^{320} T^{20} \) |
| 5 | \( 1 - \)\(32\!\cdots\!46\)\( p^{2} T^{2} + \)\(26\!\cdots\!53\)\( p^{6} T^{4} - \)\(63\!\cdots\!08\)\( p^{12} T^{6} + \)\(98\!\cdots\!26\)\( p^{21} T^{8} - \)\(25\!\cdots\!36\)\( p^{31} T^{10} + \)\(98\!\cdots\!26\)\( p^{85} T^{12} - \)\(63\!\cdots\!08\)\( p^{140} T^{14} + \)\(26\!\cdots\!53\)\( p^{198} T^{16} - \)\(32\!\cdots\!46\)\( p^{258} T^{18} + p^{320} T^{20} \) | |
| 7 | \( ( 1 + 397718744210 p T + \)\(49\!\cdots\!63\)\( p T^{2} + \)\(12\!\cdots\!60\)\( p^{3} T^{3} + \)\(52\!\cdots\!82\)\( p^{6} T^{4} + \)\(37\!\cdots\!80\)\( p^{10} T^{5} + \)\(52\!\cdots\!82\)\( p^{38} T^{6} + \)\(12\!\cdots\!60\)\( p^{67} T^{7} + \)\(49\!\cdots\!63\)\( p^{97} T^{8} + 397718744210 p^{129} T^{9} + p^{160} T^{10} )^{2} \) | |
| 11 | \( 1 - \)\(69\!\cdots\!70\)\( p T^{2} + \)\(27\!\cdots\!25\)\( p^{2} T^{4} - \)\(82\!\cdots\!00\)\( p^{3} T^{6} + \)\(15\!\cdots\!70\)\( p^{7} T^{8} - \)\(24\!\cdots\!32\)\( p^{11} T^{10} + \)\(15\!\cdots\!70\)\( p^{71} T^{12} - \)\(82\!\cdots\!00\)\( p^{131} T^{14} + \)\(27\!\cdots\!25\)\( p^{194} T^{16} - \)\(69\!\cdots\!70\)\( p^{257} T^{18} + p^{320} T^{20} \) | |
| 13 | \( ( 1 - 283848778839902890 T + \)\(96\!\cdots\!69\)\( p^{2} T^{2} - \)\(15\!\cdots\!60\)\( p^{2} T^{3} + \)\(42\!\cdots\!38\)\( p^{4} T^{4} - \)\(28\!\cdots\!00\)\( p^{6} T^{5} + \)\(42\!\cdots\!38\)\( p^{36} T^{6} - \)\(15\!\cdots\!60\)\( p^{66} T^{7} + \)\(96\!\cdots\!69\)\( p^{98} T^{8} - 283848778839902890 p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 17 | \( 1 - \)\(18\!\cdots\!38\)\( T^{2} + \)\(17\!\cdots\!77\)\( T^{4} - \)\(33\!\cdots\!52\)\( p^{2} T^{6} + \)\(44\!\cdots\!02\)\( p^{4} T^{8} - \)\(42\!\cdots\!32\)\( p^{6} T^{10} + \)\(44\!\cdots\!02\)\( p^{68} T^{12} - \)\(33\!\cdots\!52\)\( p^{130} T^{14} + \)\(17\!\cdots\!77\)\( p^{192} T^{16} - \)\(18\!\cdots\!38\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 19 | \( ( 1 - \)\(28\!\cdots\!38\)\( T + \)\(24\!\cdots\!97\)\( T^{2} - \)\(25\!\cdots\!72\)\( p T^{3} + \)\(91\!\cdots\!42\)\( p^{2} T^{4} - \)\(84\!\cdots\!12\)\( p^{3} T^{5} + \)\(91\!\cdots\!42\)\( p^{34} T^{6} - \)\(25\!\cdots\!72\)\( p^{65} T^{7} + \)\(24\!\cdots\!97\)\( p^{96} T^{8} - \)\(28\!\cdots\!38\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 23 | \( 1 - \)\(14\!\cdots\!38\)\( T^{2} + \)\(15\!\cdots\!53\)\( p^{2} T^{4} - \)\(64\!\cdots\!28\)\( p^{4} T^{6} - \)\(17\!\cdots\!82\)\( p^{6} T^{8} + \)\(33\!\cdots\!32\)\( p^{8} T^{10} - \)\(17\!\cdots\!82\)\( p^{70} T^{12} - \)\(64\!\cdots\!28\)\( p^{132} T^{14} + \)\(15\!\cdots\!53\)\( p^{194} T^{16} - \)\(14\!\cdots\!38\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 29 | \( 1 - \)\(48\!\cdots\!50\)\( p^{2} T^{2} + \)\(11\!\cdots\!65\)\( p^{4} T^{4} - \)\(17\!\cdots\!40\)\( p^{6} T^{6} + \)\(19\!\cdots\!50\)\( p^{8} T^{8} - \)\(16\!\cdots\!52\)\( p^{10} T^{10} + \)\(19\!\cdots\!50\)\( p^{72} T^{12} - \)\(17\!\cdots\!40\)\( p^{134} T^{14} + \)\(11\!\cdots\!65\)\( p^{196} T^{16} - \)\(48\!\cdots\!50\)\( p^{258} T^{18} + p^{320} T^{20} \) | |
| 31 | \( ( 1 - \)\(37\!\cdots\!70\)\( p T + \)\(21\!\cdots\!25\)\( p^{2} T^{2} - \)\(40\!\cdots\!00\)\( p^{3} T^{3} + \)\(14\!\cdots\!70\)\( p^{4} T^{4} - \)\(19\!\cdots\!52\)\( p^{5} T^{5} + \)\(14\!\cdots\!70\)\( p^{36} T^{6} - \)\(40\!\cdots\!00\)\( p^{67} T^{7} + \)\(21\!\cdots\!25\)\( p^{98} T^{8} - \)\(37\!\cdots\!70\)\( p^{129} T^{9} + p^{160} T^{10} )^{2} \) | |
| 37 | \( ( 1 + \)\(17\!\cdots\!70\)\( T + \)\(55\!\cdots\!61\)\( T^{2} + \)\(85\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!18\)\( p^{32} T^{6} + \)\(85\!\cdots\!60\)\( p^{64} T^{7} + \)\(55\!\cdots\!61\)\( p^{96} T^{8} + \)\(17\!\cdots\!70\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 41 | \( 1 - \)\(30\!\cdots\!70\)\( T^{2} + \)\(43\!\cdots\!25\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!70\)\( p^{64} T^{12} - \)\(40\!\cdots\!00\)\( p^{128} T^{14} + \)\(43\!\cdots\!25\)\( p^{192} T^{16} - \)\(30\!\cdots\!70\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 43 | \( ( 1 - \)\(10\!\cdots\!70\)\( T + \)\(53\!\cdots\!61\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(39\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!78\)\( p^{32} T^{6} + \)\(13\!\cdots\!20\)\( p^{64} T^{7} + \)\(53\!\cdots\!61\)\( p^{96} T^{8} - \)\(10\!\cdots\!70\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 47 | \( 1 - \)\(12\!\cdots\!38\)\( T^{2} + \)\(90\!\cdots\!97\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{6} + \)\(20\!\cdots\!82\)\( T^{8} - \)\(70\!\cdots\!68\)\( T^{10} + \)\(20\!\cdots\!82\)\( p^{64} T^{12} - \)\(47\!\cdots\!08\)\( p^{128} T^{14} + \)\(90\!\cdots\!97\)\( p^{192} T^{16} - \)\(12\!\cdots\!38\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 53 | \( 1 - \)\(91\!\cdots\!38\)\( T^{2} + \)\(41\!\cdots\!97\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + \)\(26\!\cdots\!82\)\( T^{8} - \)\(15\!\cdots\!52\)\( p^{2} T^{10} + \)\(26\!\cdots\!82\)\( p^{64} T^{12} - \)\(12\!\cdots\!08\)\( p^{128} T^{14} + \)\(41\!\cdots\!97\)\( p^{192} T^{16} - \)\(91\!\cdots\!38\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 59 | \( 1 - \)\(20\!\cdots\!70\)\( T^{2} + \)\(24\!\cdots\!25\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} - \)\(67\!\cdots\!52\)\( T^{10} + \)\(13\!\cdots\!70\)\( p^{64} T^{12} - \)\(20\!\cdots\!00\)\( p^{128} T^{14} + \)\(24\!\cdots\!25\)\( p^{192} T^{16} - \)\(20\!\cdots\!70\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 61 | \( ( 1 + \)\(98\!\cdots\!90\)\( T + \)\(77\!\cdots\!45\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!10\)\( T^{4} + \)\(87\!\cdots\!48\)\( T^{5} + \)\(22\!\cdots\!10\)\( p^{32} T^{6} + \)\(45\!\cdots\!80\)\( p^{64} T^{7} + \)\(77\!\cdots\!45\)\( p^{96} T^{8} + \)\(98\!\cdots\!90\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 67 | \( ( 1 + \)\(22\!\cdots\!70\)\( T + \)\(81\!\cdots\!43\)\( p T^{2} + \)\(38\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!18\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!18\)\( p^{32} T^{6} + \)\(38\!\cdots\!40\)\( p^{64} T^{7} + \)\(81\!\cdots\!43\)\( p^{97} T^{8} + \)\(22\!\cdots\!70\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 71 | \( 1 - \)\(55\!\cdots\!50\)\( T^{2} + \)\(19\!\cdots\!65\)\( T^{4} - \)\(53\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(21\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!50\)\( p^{64} T^{12} - \)\(53\!\cdots\!40\)\( p^{128} T^{14} + \)\(19\!\cdots\!65\)\( p^{192} T^{16} - \)\(55\!\cdots\!50\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 73 | \( ( 1 + \)\(22\!\cdots\!10\)\( T + \)\(38\!\cdots\!81\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!98\)\( T^{4} + \)\(28\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!98\)\( p^{32} T^{6} + \)\(43\!\cdots\!20\)\( p^{64} T^{7} + \)\(38\!\cdots\!81\)\( p^{96} T^{8} + \)\(22\!\cdots\!10\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 79 | \( ( 1 + \)\(10\!\cdots\!82\)\( T + \)\(11\!\cdots\!17\)\( T^{2} + \)\(49\!\cdots\!32\)\( T^{3} + \)\(61\!\cdots\!22\)\( T^{4} + \)\(79\!\cdots\!48\)\( p T^{5} + \)\(61\!\cdots\!22\)\( p^{32} T^{6} + \)\(49\!\cdots\!32\)\( p^{64} T^{7} + \)\(11\!\cdots\!17\)\( p^{96} T^{8} + \)\(10\!\cdots\!82\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 83 | \( 1 - \)\(14\!\cdots\!98\)\( T^{2} + \)\(78\!\cdots\!77\)\( T^{4} - \)\(15\!\cdots\!88\)\( T^{6} - \)\(27\!\cdots\!58\)\( T^{8} + \)\(18\!\cdots\!32\)\( T^{10} - \)\(27\!\cdots\!58\)\( p^{64} T^{12} - \)\(15\!\cdots\!88\)\( p^{128} T^{14} + \)\(78\!\cdots\!77\)\( p^{192} T^{16} - \)\(14\!\cdots\!98\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 89 | \( 1 - \)\(19\!\cdots\!70\)\( T^{2} + \)\(17\!\cdots\!25\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!70\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(37\!\cdots\!70\)\( p^{64} T^{12} - \)\(96\!\cdots\!00\)\( p^{128} T^{14} + \)\(17\!\cdots\!25\)\( p^{192} T^{16} - \)\(19\!\cdots\!70\)\( p^{256} T^{18} + p^{320} T^{20} \) | |
| 97 | \( ( 1 - \)\(19\!\cdots\!50\)\( T + \)\(26\!\cdots\!61\)\( T^{2} - \)\(24\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!98\)\( p^{32} T^{6} - \)\(24\!\cdots\!60\)\( p^{64} T^{7} + \)\(26\!\cdots\!61\)\( p^{96} T^{8} - \)\(19\!\cdots\!50\)\( p^{128} T^{9} + p^{160} T^{10} )^{2} \) | |
| 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
−4.88236418382632927017132758345, −4.85313163604371770114396938655, −4.81827932898750655942509005667, −4.75048784892673222655195664649, −4.46506279558335241309862528385, −4.24031201318498397317921863752, −3.63507821602921171877871336480, −3.56923892365925808228323183050, −3.20192629420040226426676520396, −3.08279953417597026079766488255, −2.94573668541317151540832919514, −2.93334866743795456422373880043, −2.82213356571156191330043390629, −2.80957839905058613594330131953, −2.03084365240619974629838488881, −1.89926867162753171766414309590, −1.54345496345849636242513314267, −1.54102811666769356088107745088, −1.51383087727851512689728916636, −1.39007541162590128601111677727, −1.11037056208000841333666017356, −0.73280820389272097092063334738, −0.47066966545282972853746542975, −0.44945373501547614742774698264, −0.10339536107275742927649452116, 0.10339536107275742927649452116, 0.44945373501547614742774698264, 0.47066966545282972853746542975, 0.73280820389272097092063334738, 1.11037056208000841333666017356, 1.39007541162590128601111677727, 1.51383087727851512689728916636, 1.54102811666769356088107745088, 1.54345496345849636242513314267, 1.89926867162753171766414309590, 2.03084365240619974629838488881, 2.80957839905058613594330131953, 2.82213356571156191330043390629, 2.93334866743795456422373880043, 2.94573668541317151540832919514, 3.08279953417597026079766488255, 3.20192629420040226426676520396, 3.56923892365925808228323183050, 3.63507821602921171877871336480, 4.24031201318498397317921863752, 4.46506279558335241309862528385, 4.75048784892673222655195664649, 4.81827932898750655942509005667, 4.85313163604371770114396938655, 4.88236418382632927017132758345
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.