Dirichlet series
| L(s) = 1 | − 2.33e3·7-s + 3.75e4·11-s + 2.23e4·23-s + 1.80e6·25-s + 3.08e5·29-s − 5.47e6·37-s + 1.17e6·43-s − 2.98e5·49-s + 1.29e5·53-s − 5.72e6·67-s − 2.69e7·71-s − 8.78e7·77-s − 1.81e8·79-s + 7.40e8·107-s − 2.57e8·109-s − 5.11e8·113-s − 8.42e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 5.23e7·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.973·7-s + 2.56·11-s + 0.0799·23-s + 4.61·25-s + 0.436·29-s − 2.91·37-s + 0.344·43-s − 0.0518·49-s + 0.0163·53-s − 0.283·67-s − 1.06·71-s − 2.50·77-s − 4.65·79-s + 5.64·107-s − 1.82·109-s − 3.13·113-s − 3.93·121-s − 0.0778·161-s + 4.66·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+4)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 7^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.30013\times 10^{20}\) |
| Root analytic conductor: | \(10.1320\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 2^{20} \cdot 3^{20} \cdot 7^{10} ,\ ( \ : [4]^{10} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(70.25093749\) |
| \(L(\frac12)\) | \(\approx\) | \(70.25093749\) |
| \(L(5)\) | 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 | \( 1 \) | |
| 7 | \( 1 + 334 p T + 117657 p^{2} T^{2} + 174576 p^{6} T^{3} + 222529494 p^{6} T^{4} + 172560924 p^{10} T^{5} + 222529494 p^{14} T^{6} + 174576 p^{22} T^{7} + 117657 p^{26} T^{8} + 334 p^{33} T^{9} + p^{40} T^{10} \) | |
| good | 5 | \( 1 - 1801126 T^{2} + 1654569812049 T^{4} - 1045330593107636832 T^{6} + \)\(42\!\cdots\!26\)\( p^{3} T^{8} - \)\(35\!\cdots\!32\)\( p^{4} T^{10} + \)\(42\!\cdots\!26\)\( p^{19} T^{12} - 1045330593107636832 p^{32} T^{14} + 1654569812049 p^{48} T^{16} - 1801126 p^{64} T^{18} + p^{80} T^{20} \) |
| 11 | \( ( 1 - 18798 T + 951398579 T^{2} - 14346554158212 T^{3} + 394798313359326220 T^{4} - \)\(44\!\cdots\!60\)\( T^{5} + 394798313359326220 p^{8} T^{6} - 14346554158212 p^{16} T^{7} + 951398579 p^{24} T^{8} - 18798 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 13 | \( 1 - 3808243978 T^{2} + 6683878414560511917 T^{4} - \)\(80\!\cdots\!68\)\( T^{6} + \)\(84\!\cdots\!06\)\( T^{8} - \)\(75\!\cdots\!28\)\( T^{10} + \)\(84\!\cdots\!06\)\( p^{16} T^{12} - \)\(80\!\cdots\!68\)\( p^{32} T^{14} + 6683878414560511917 p^{48} T^{16} - 3808243978 p^{64} T^{18} + p^{80} T^{20} \) | |
| 17 | \( 1 - 36719622742 T^{2} + \)\(74\!\cdots\!93\)\( T^{4} - \)\(10\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!94\)\( T^{8} - \)\(80\!\cdots\!00\)\( T^{10} + \)\(10\!\cdots\!94\)\( p^{16} T^{12} - \)\(10\!\cdots\!52\)\( p^{32} T^{14} + \)\(74\!\cdots\!93\)\( p^{48} T^{16} - 36719622742 p^{64} T^{18} + p^{80} T^{20} \) | |
| 19 | \( 1 - 76037725090 T^{2} + \)\(31\!\cdots\!33\)\( T^{4} - \)\(94\!\cdots\!76\)\( T^{6} + \)\(22\!\cdots\!34\)\( T^{8} - \)\(41\!\cdots\!32\)\( T^{10} + \)\(22\!\cdots\!34\)\( p^{16} T^{12} - \)\(94\!\cdots\!76\)\( p^{32} T^{14} + \)\(31\!\cdots\!33\)\( p^{48} T^{16} - 76037725090 p^{64} T^{18} + p^{80} T^{20} \) | |
| 23 | \( ( 1 - 11190 T + 126500339699 T^{2} + 6299007423824508 T^{3} + \)\(16\!\cdots\!52\)\( T^{4} + \)\(12\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!52\)\( p^{8} T^{6} + 6299007423824508 p^{16} T^{7} + 126500339699 p^{24} T^{8} - 11190 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 29 | \( ( 1 - 154446 T + 1138790023229 T^{2} - 397550561533808136 T^{3} + \)\(78\!\cdots\!14\)\( T^{4} - \)\(31\!\cdots\!28\)\( T^{5} + \)\(78\!\cdots\!14\)\( p^{8} T^{6} - 397550561533808136 p^{16} T^{7} + 1138790023229 p^{24} T^{8} - 154446 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 31 | \( 1 - 5199410804650 T^{2} + \)\(11\!\cdots\!37\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{6} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(84\!\cdots\!40\)\( T^{10} + \)\(11\!\cdots\!22\)\( p^{16} T^{12} - \)\(14\!\cdots\!44\)\( p^{32} T^{14} + \)\(11\!\cdots\!37\)\( p^{48} T^{16} - 5199410804650 p^{64} T^{18} + p^{80} T^{20} \) | |
| 37 | \( ( 1 + 2735554 T + 9873113449713 T^{2} + 7881485694642506688 T^{3} + \)\(12\!\cdots\!54\)\( T^{4} - \)\(20\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!54\)\( p^{8} T^{6} + 7881485694642506688 p^{16} T^{7} + 9873113449713 p^{24} T^{8} + 2735554 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 41 | \( 1 - 34206366782710 T^{2} + \)\(63\!\cdots\!09\)\( T^{4} - \)\(86\!\cdots\!96\)\( T^{6} + \)\(91\!\cdots\!82\)\( T^{8} - \)\(79\!\cdots\!52\)\( T^{10} + \)\(91\!\cdots\!82\)\( p^{16} T^{12} - \)\(86\!\cdots\!96\)\( p^{32} T^{14} + \)\(63\!\cdots\!09\)\( p^{48} T^{16} - 34206366782710 p^{64} T^{18} + p^{80} T^{20} \) | |
| 43 | \( ( 1 - 588662 T + 26723095003977 T^{2} - 22492698026016018864 T^{3} + \)\(44\!\cdots\!58\)\( T^{4} - \)\(52\!\cdots\!56\)\( T^{5} + \)\(44\!\cdots\!58\)\( p^{8} T^{6} - 22492698026016018864 p^{16} T^{7} + 26723095003977 p^{24} T^{8} - 588662 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 47 | \( 1 - 40671321614314 T^{2} + \)\(47\!\cdots\!85\)\( T^{4} - \)\(97\!\cdots\!08\)\( p T^{6} + \)\(60\!\cdots\!50\)\( T^{8} - \)\(25\!\cdots\!44\)\( T^{10} + \)\(60\!\cdots\!50\)\( p^{16} T^{12} - \)\(97\!\cdots\!08\)\( p^{33} T^{14} + \)\(47\!\cdots\!85\)\( p^{48} T^{16} - 40671321614314 p^{64} T^{18} + p^{80} T^{20} \) | |
| 53 | \( ( 1 - 64566 T + 109877569098701 T^{2} - \)\(14\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{5} + \)\(10\!\cdots\!86\)\( p^{8} T^{6} - \)\(14\!\cdots\!88\)\( p^{16} T^{7} + 109877569098701 p^{24} T^{8} - 64566 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 59 | \( 1 - 719850081974554 T^{2} + \)\(27\!\cdots\!33\)\( T^{4} - \)\(74\!\cdots\!44\)\( T^{6} + \)\(15\!\cdots\!78\)\( T^{8} - \)\(24\!\cdots\!40\)\( T^{10} + \)\(15\!\cdots\!78\)\( p^{16} T^{12} - \)\(74\!\cdots\!44\)\( p^{32} T^{14} + \)\(27\!\cdots\!33\)\( p^{48} T^{16} - 719850081974554 p^{64} T^{18} + p^{80} T^{20} \) | |
| 61 | \( 1 - 879605222098522 T^{2} + \)\(36\!\cdots\!45\)\( T^{4} - \)\(92\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!90\)\( T^{8} - \)\(29\!\cdots\!96\)\( T^{10} + \)\(16\!\cdots\!90\)\( p^{16} T^{12} - \)\(92\!\cdots\!60\)\( p^{32} T^{14} + \)\(36\!\cdots\!45\)\( p^{48} T^{16} - 879605222098522 p^{64} T^{18} + p^{80} T^{20} \) | |
| 67 | \( ( 1 + 2861186 T + 330082766774289 T^{2} + \)\(72\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!10\)\( T^{4} + \)\(61\!\cdots\!12\)\( T^{5} + \)\(30\!\cdots\!10\)\( p^{8} T^{6} + \)\(72\!\cdots\!28\)\( p^{16} T^{7} + 330082766774289 p^{24} T^{8} + 2861186 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 71 | \( ( 1 + 13492770 T + 2606256464420867 T^{2} + \)\(29\!\cdots\!48\)\( T^{3} + \)\(30\!\cdots\!20\)\( T^{4} + \)\(26\!\cdots\!68\)\( T^{5} + \)\(30\!\cdots\!20\)\( p^{8} T^{6} + \)\(29\!\cdots\!48\)\( p^{16} T^{7} + 2606256464420867 p^{24} T^{8} + 13492770 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 73 | \( 1 - 6341686899316858 T^{2} + \)\(18\!\cdots\!45\)\( T^{4} - \)\(35\!\cdots\!68\)\( T^{6} + \)\(45\!\cdots\!50\)\( T^{8} - \)\(42\!\cdots\!60\)\( T^{10} + \)\(45\!\cdots\!50\)\( p^{16} T^{12} - \)\(35\!\cdots\!68\)\( p^{32} T^{14} + \)\(18\!\cdots\!45\)\( p^{48} T^{16} - 6341686899316858 p^{64} T^{18} + p^{80} T^{20} \) | |
| 79 | \( ( 1 + 90598778 T + 8046771666605457 T^{2} + \)\(42\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!38\)\( T^{4} + \)\(85\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!38\)\( p^{8} T^{6} + \)\(42\!\cdots\!12\)\( p^{16} T^{7} + 8046771666605457 p^{24} T^{8} + 90598778 p^{32} T^{9} + p^{40} T^{10} )^{2} \) | |
| 83 | \( 1 - 19278785649623290 T^{2} + \)\(17\!\cdots\!85\)\( T^{4} - \)\(95\!\cdots\!28\)\( T^{6} + \)\(35\!\cdots\!42\)\( T^{8} - \)\(95\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!42\)\( p^{16} T^{12} - \)\(95\!\cdots\!28\)\( p^{32} T^{14} + \)\(17\!\cdots\!85\)\( p^{48} T^{16} - 19278785649623290 p^{64} T^{18} + p^{80} T^{20} \) | |
| 89 | \( 1 - 28192651452869014 T^{2} + \)\(36\!\cdots\!61\)\( T^{4} - \)\(29\!\cdots\!64\)\( T^{6} + \)\(17\!\cdots\!06\)\( T^{8} - \)\(75\!\cdots\!80\)\( T^{10} + \)\(17\!\cdots\!06\)\( p^{16} T^{12} - \)\(29\!\cdots\!64\)\( p^{32} T^{14} + \)\(36\!\cdots\!61\)\( p^{48} T^{16} - 28192651452869014 p^{64} T^{18} + p^{80} T^{20} \) | |
| 97 | \( 1 - 70135963279762234 T^{2} + \)\(22\!\cdots\!17\)\( T^{4} - \)\(44\!\cdots\!04\)\( T^{6} + \)\(60\!\cdots\!78\)\( p T^{8} - \)\(54\!\cdots\!80\)\( T^{10} + \)\(60\!\cdots\!78\)\( p^{17} T^{12} - \)\(44\!\cdots\!04\)\( p^{32} T^{14} + \)\(22\!\cdots\!17\)\( p^{48} T^{16} - 70135963279762234 p^{64} T^{18} + p^{80} T^{20} \) | |
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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
−3.21670516481154718752167765957, −3.16830466536225787175846964893, −3.03862389457153237157468075399, −2.95155328580253839359918339424, −2.67679118729125045170126430885, −2.57358860101295428783252626609, −2.56395256792685622195629346949, −2.42254089222897478084182593499, −2.22777023423081566823790268043, −1.98764427479838984074911140217, −1.79046517083128379681652593421, −1.66139711101016137842885020401, −1.65908074355797062174751624111, −1.49216885378794838821131538181, −1.40497673363973236338315403153, −1.36865833564717145674856904974, −1.27316070573104311789264489113, −1.03324897832113609116950078097, −0.75206890247051474845287136309, −0.71532861163780227611360499831, −0.55296546447058148020817538985, −0.50434979226513152010980738462, −0.47116182001538848507232076734, −0.30147812514827534527206283393, −0.23355501312277267329648222659, 0.23355501312277267329648222659, 0.30147812514827534527206283393, 0.47116182001538848507232076734, 0.50434979226513152010980738462, 0.55296546447058148020817538985, 0.71532861163780227611360499831, 0.75206890247051474845287136309, 1.03324897832113609116950078097, 1.27316070573104311789264489113, 1.36865833564717145674856904974, 1.40497673363973236338315403153, 1.49216885378794838821131538181, 1.65908074355797062174751624111, 1.66139711101016137842885020401, 1.79046517083128379681652593421, 1.98764427479838984074911140217, 2.22777023423081566823790268043, 2.42254089222897478084182593499, 2.56395256792685622195629346949, 2.57358860101295428783252626609, 2.67679118729125045170126430885, 2.95155328580253839359918339424, 3.03862389457153237157468075399, 3.16830466536225787175846964893, 3.21670516481154718752167765957
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.