Dirichlet series
| L(s) = 1 | − 22·3-s − 5.43e3·7-s − 1.42e4·9-s − 1.24e5·13-s − 4.89e6·19-s + 1.19e5·21-s + 3.99e7·25-s − 8.62e5·27-s − 4.25e6·31-s + 8.99e7·37-s + 2.73e6·39-s + 1.59e8·43-s − 1.24e9·49-s + 1.07e8·57-s − 1.01e8·61-s + 7.73e7·63-s − 3.01e9·67-s + 4.92e9·73-s − 8.79e8·75-s − 7.63e9·79-s − 1.79e9·81-s + 6.76e8·91-s + 9.35e7·93-s + 3.78e10·97-s − 5.51e10·103-s + 6.51e10·109-s − 1.97e9·111-s + ⋯ |
| L(s) = 1 | − 0.0905·3-s − 0.323·7-s − 0.240·9-s − 0.335·13-s − 1.97·19-s + 0.0292·21-s + 4.09·25-s − 0.0600·27-s − 0.148·31-s + 1.29·37-s + 0.0303·39-s + 1.08·43-s − 4.41·49-s + 0.178·57-s − 0.120·61-s + 0.0779·63-s − 2.23·67-s + 2.37·73-s − 0.370·75-s − 2.48·79-s − 0.515·81-s + 0.108·91-s + 0.0134·93-s + 4.40·97-s − 4.76·103-s + 4.23·109-s − 0.117·111-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{10}\right)^{s/2} \, \Gamma_{\C}(s+5)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{30} \cdot 3^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.79660\times 10^{11}\) |
| Root analytic conductor: | \(3.90494\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{30} \cdot 3^{10} ,\ ( \ : [5]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(4.286113409\) |
| \(L(\frac12)\) | \(\approx\) | \(4.286113409\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 22 T + 4903 p T^{2} + 2056 p^{6} T^{3} + 104614 p^{9} T^{4} + 15708004 p^{10} T^{5} + 104614 p^{19} T^{6} + 2056 p^{26} T^{7} + 4903 p^{31} T^{8} + 22 p^{40} T^{9} + p^{50} T^{10} \) | |
| good | 5 | \( 1 - 39957018 T^{2} + 152190104080057 p T^{4} - \)\(35\!\cdots\!68\)\( p^{2} T^{6} + \)\(60\!\cdots\!62\)\( p^{3} T^{8} - \)\(10\!\cdots\!88\)\( p^{4} T^{10} + \)\(60\!\cdots\!62\)\( p^{23} T^{12} - \)\(35\!\cdots\!68\)\( p^{42} T^{14} + 152190104080057 p^{61} T^{16} - 39957018 p^{80} T^{18} + p^{100} T^{20} \) |
| 7 | \( ( 1 + 2718 T + 634511645 T^{2} + 991398861144 p T^{3} + 4925007459309682 p^{2} T^{4} + 8085900600584262828 p^{3} T^{5} + 4925007459309682 p^{12} T^{6} + 991398861144 p^{21} T^{7} + 634511645 p^{30} T^{8} + 2718 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 11 | \( 1 - 193963482330 T^{2} + \)\(17\!\cdots\!69\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{10} + \)\(42\!\cdots\!86\)\( p^{20} T^{12} - \)\(10\!\cdots\!60\)\( p^{40} T^{14} + \)\(17\!\cdots\!69\)\( p^{60} T^{16} - 193963482330 p^{80} T^{18} + p^{100} T^{20} \) | |
| 13 | \( ( 1 + 62254 T + 278498841045 T^{2} + 17676468508575336 T^{3} + \)\(29\!\cdots\!94\)\( T^{4} + \)\(31\!\cdots\!60\)\( T^{5} + \)\(29\!\cdots\!94\)\( p^{10} T^{6} + 17676468508575336 p^{20} T^{7} + 278498841045 p^{30} T^{8} + 62254 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 17 | \( 1 - 9256904773194 T^{2} + \)\(49\!\cdots\!33\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(53\!\cdots\!94\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{10} + \)\(53\!\cdots\!94\)\( p^{20} T^{12} - \)\(18\!\cdots\!12\)\( p^{40} T^{14} + \)\(49\!\cdots\!33\)\( p^{60} T^{16} - 9256904773194 p^{80} T^{18} + p^{100} T^{20} \) | |
| 19 | \( ( 1 + 2446742 T + 11680332608469 T^{2} + 40888201556594279880 T^{3} + \)\(64\!\cdots\!74\)\( p T^{4} + \)\(82\!\cdots\!04\)\( p^{2} T^{5} + \)\(64\!\cdots\!74\)\( p^{11} T^{6} + 40888201556594279880 p^{20} T^{7} + 11680332608469 p^{30} T^{8} + 2446742 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 23 | \( 1 - 107664482004138 T^{2} + \)\(11\!\cdots\!85\)\( T^{4} - \)\(72\!\cdots\!88\)\( T^{6} + \)\(43\!\cdots\!94\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!94\)\( p^{20} T^{12} - \)\(72\!\cdots\!88\)\( p^{40} T^{14} + \)\(11\!\cdots\!85\)\( p^{60} T^{16} - 107664482004138 p^{80} T^{18} + p^{100} T^{20} \) | |
| 29 | \( 1 - 2079099253535866 T^{2} + \)\(23\!\cdots\!93\)\( T^{4} - \)\(18\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!74\)\( T^{8} - \)\(51\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!74\)\( p^{20} T^{12} - \)\(18\!\cdots\!08\)\( p^{40} T^{14} + \)\(23\!\cdots\!93\)\( p^{60} T^{16} - 2079099253535866 p^{80} T^{18} + p^{100} T^{20} \) | |
| 31 | \( ( 1 + 2125742 T + 3378299019643821 T^{2} + \)\(53\!\cdots\!80\)\( T^{3} + \)\(50\!\cdots\!82\)\( T^{4} + \)\(62\!\cdots\!84\)\( T^{5} + \)\(50\!\cdots\!82\)\( p^{10} T^{6} + \)\(53\!\cdots\!80\)\( p^{20} T^{7} + 3378299019643821 p^{30} T^{8} + 2125742 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 37 | \( ( 1 - 44992578 T + 7273564877478533 T^{2} - \)\(50\!\cdots\!52\)\( T^{3} + \)\(37\!\cdots\!86\)\( T^{4} - \)\(36\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!86\)\( p^{10} T^{6} - \)\(50\!\cdots\!52\)\( p^{20} T^{7} + 7273564877478533 p^{30} T^{8} - 44992578 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 41 | \( 1 - 28773388104915370 T^{2} + \)\(95\!\cdots\!25\)\( T^{4} - \)\(44\!\cdots\!80\)\( p T^{6} + \)\(35\!\cdots\!70\)\( T^{8} - \)\(47\!\cdots\!20\)\( T^{10} + \)\(35\!\cdots\!70\)\( p^{20} T^{12} - \)\(44\!\cdots\!80\)\( p^{41} T^{14} + \)\(95\!\cdots\!25\)\( p^{60} T^{16} - 28773388104915370 p^{80} T^{18} + p^{100} T^{20} \) | |
| 43 | \( ( 1 - 79993658 T + 49655088622021029 T^{2} - \)\(44\!\cdots\!32\)\( p T^{3} + \)\(12\!\cdots\!62\)\( T^{4} - \)\(70\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!62\)\( p^{10} T^{6} - \)\(44\!\cdots\!32\)\( p^{21} T^{7} + 49655088622021029 p^{30} T^{8} - 79993658 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 47 | \( 1 - 69441444227446090 T^{2} + \)\(13\!\cdots\!45\)\( T^{4} - \)\(68\!\cdots\!60\)\( T^{6} + \)\(69\!\cdots\!30\)\( T^{8} - \)\(27\!\cdots\!40\)\( T^{10} + \)\(69\!\cdots\!30\)\( p^{20} T^{12} - \)\(68\!\cdots\!60\)\( p^{40} T^{14} + \)\(13\!\cdots\!45\)\( p^{60} T^{16} - 69441444227446090 p^{80} T^{18} + p^{100} T^{20} \) | |
| 53 | \( 1 - 921576921816876762 T^{2} + \)\(46\!\cdots\!97\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!38\)\( T^{8} - \)\(79\!\cdots\!12\)\( T^{10} + \)\(40\!\cdots\!38\)\( p^{20} T^{12} - \)\(15\!\cdots\!20\)\( p^{40} T^{14} + \)\(46\!\cdots\!97\)\( p^{60} T^{16} - 921576921816876762 p^{80} T^{18} + p^{100} T^{20} \) | |
| 59 | \( 1 - 161151983244682266 T^{2} + \)\(33\!\cdots\!77\)\( T^{4} + \)\(10\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!34\)\( T^{8} + \)\(18\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!34\)\( p^{20} T^{12} + \)\(10\!\cdots\!52\)\( p^{40} T^{14} + \)\(33\!\cdots\!77\)\( p^{60} T^{16} - 161151983244682266 p^{80} T^{18} + p^{100} T^{20} \) | |
| 61 | \( ( 1 + 50730382 T + 2432185295050664373 T^{2} - \)\(51\!\cdots\!96\)\( T^{3} + \)\(28\!\cdots\!14\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!14\)\( p^{10} T^{6} - \)\(51\!\cdots\!96\)\( p^{20} T^{7} + 2432185295050664373 p^{30} T^{8} + 50730382 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 67 | \( ( 1 + 1507264022 T + 6221376457534648053 T^{2} + \)\(87\!\cdots\!04\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} + \)\(21\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!54\)\( p^{10} T^{6} + \)\(87\!\cdots\!04\)\( p^{20} T^{7} + 6221376457534648053 p^{30} T^{8} + 1507264022 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 71 | \( 1 - 12553452108092223978 T^{2} + \)\(92\!\cdots\!33\)\( T^{4} - \)\(49\!\cdots\!76\)\( T^{6} + \)\(30\!\cdots\!14\)\( p T^{8} - \)\(78\!\cdots\!00\)\( T^{10} + \)\(30\!\cdots\!14\)\( p^{21} T^{12} - \)\(49\!\cdots\!76\)\( p^{40} T^{14} + \)\(92\!\cdots\!33\)\( p^{60} T^{16} - 12553452108092223978 p^{80} T^{18} + p^{100} T^{20} \) | |
| 73 | \( ( 1 - 2460461018 T + 19102782343327863549 T^{2} - \)\(36\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!22\)\( T^{4} - \)\(22\!\cdots\!04\)\( T^{5} + \)\(15\!\cdots\!22\)\( p^{10} T^{6} - \)\(36\!\cdots\!36\)\( p^{20} T^{7} + 19102782343327863549 p^{30} T^{8} - 2460461018 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 79 | \( ( 1 + 3815845006 T + 28661695473434414157 T^{2} + \)\(97\!\cdots\!28\)\( T^{3} + \)\(43\!\cdots\!82\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{5} + \)\(43\!\cdots\!82\)\( p^{10} T^{6} + \)\(97\!\cdots\!28\)\( p^{20} T^{7} + 28661695473434414157 p^{30} T^{8} + 3815845006 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
| 83 | \( 1 - 720890427493020702 p T^{2} + \)\(22\!\cdots\!77\)\( T^{4} - \)\(60\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!62\)\( T^{8} - \)\(22\!\cdots\!84\)\( T^{10} + \)\(12\!\cdots\!62\)\( p^{20} T^{12} - \)\(60\!\cdots\!68\)\( p^{40} T^{14} + \)\(22\!\cdots\!77\)\( p^{60} T^{16} - 720890427493020702 p^{81} T^{18} + p^{100} T^{20} \) | |
| 89 | \( 1 - \)\(15\!\cdots\!46\)\( T^{2} + \)\(13\!\cdots\!61\)\( T^{4} - \)\(80\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!14\)\( T^{8} - \)\(12\!\cdots\!28\)\( T^{10} + \)\(36\!\cdots\!14\)\( p^{20} T^{12} - \)\(80\!\cdots\!00\)\( p^{40} T^{14} + \)\(13\!\cdots\!61\)\( p^{60} T^{16} - \)\(15\!\cdots\!46\)\( p^{80} T^{18} + p^{100} T^{20} \) | |
| 97 | \( ( 1 - 18930689578 T + \)\(31\!\cdots\!01\)\( T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!06\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!06\)\( p^{10} T^{6} - \)\(35\!\cdots\!60\)\( p^{20} T^{7} + \)\(31\!\cdots\!01\)\( p^{30} T^{8} - 18930689578 p^{40} T^{9} + p^{50} T^{10} )^{2} \) | |
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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
−5.01869657674430812819148537388, −4.90627070610227154827857422959, −4.82169935684391035856689129203, −4.59354648420080199203540749297, −4.45256819903787175637496129889, −4.29186006574568656849530077191, −4.11681950101214450418917944662, −3.61029568740021437082854307805, −3.54772519386806316816428427858, −3.43177654732635813322042850606, −3.29325155188628339513573931843, −2.82027256007731799160170731781, −2.76450205286496982056642484464, −2.56883003704181393820810405426, −2.54783775868471452773273290465, −2.25530570777097916427553147699, −1.76038510878330574642964910440, −1.75189038263837468128448089640, −1.45827595254508359616973266621, −1.23372714163263727950969819687, −1.13031421252680724143404602544, −0.65408820295548614048252323481, −0.60105166627999763051317143929, −0.31375463201320893870754726613, −0.20660946496755309805036686348, 0.20660946496755309805036686348, 0.31375463201320893870754726613, 0.60105166627999763051317143929, 0.65408820295548614048252323481, 1.13031421252680724143404602544, 1.23372714163263727950969819687, 1.45827595254508359616973266621, 1.75189038263837468128448089640, 1.76038510878330574642964910440, 2.25530570777097916427553147699, 2.54783775868471452773273290465, 2.56883003704181393820810405426, 2.76450205286496982056642484464, 2.82027256007731799160170731781, 3.29325155188628339513573931843, 3.43177654732635813322042850606, 3.54772519386806316816428427858, 3.61029568740021437082854307805, 4.11681950101214450418917944662, 4.29186006574568656849530077191, 4.45256819903787175637496129889, 4.59354648420080199203540749297, 4.82169935684391035856689129203, 4.90627070610227154827857422959, 5.01869657674430812819148537388
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.