Dirichlet series
| L(s) = 1 | + 544·2-s − 1.04e5·4-s − 3.03e6·7-s − 1.10e8·8-s − 4.30e8·11-s − 1.65e9·14-s − 3.16e9·16-s − 2.34e11·22-s − 8.97e10·23-s + 6.41e11·25-s + 3.17e11·28-s + 2.24e10·29-s + 9.56e12·32-s + 5.73e12·37-s − 3.97e12·43-s + 4.50e13·44-s − 4.88e13·46-s − 9.12e12·49-s + 3.48e14·50-s + 1.08e14·53-s + 3.35e14·56-s + 1.22e13·58-s + 9.00e14·64-s − 7.22e14·67-s + 2.89e14·71-s + 3.12e15·74-s + 1.30e15·77-s + ⋯ |
| L(s) = 1 | + 17/8·2-s − 1.59·4-s − 0.526·7-s − 6.59·8-s − 2.00·11-s − 1.11·14-s − 0.737·16-s − 4.26·22-s − 1.14·23-s + 4.20·25-s + 0.841·28-s + 0.0448·29-s + 8.70·32-s + 1.63·37-s − 0.340·43-s + 3.21·44-s − 2.43·46-s − 0.274·49-s + 8.93·50-s + 1.74·53-s + 3.46·56-s + 0.0953·58-s + 3.19·64-s − 1.77·67-s + 0.447·71-s + 3.47·74-s + 1.05·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+8)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.19618\times 10^{16}\) |
| Root analytic conductor: | \(10.1125\) |
| Motivic weight: | \(16\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 7^{8} ,\ ( \ : [8]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{17}{2})\) | \(\approx\) | \(4.876007668\) |
| \(L(\frac12)\) | \(\approx\) | \(4.876007668\) |
| \(L(9)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 61928 p^{2} T + 1090574020 p^{5} T^{2} - 192597194408 p^{10} T^{3} - 148697277046 p^{17} T^{4} - 192597194408 p^{26} T^{5} + 1090574020 p^{37} T^{6} + 61928 p^{50} T^{7} + p^{64} T^{8} \) | |
| good | 2 | \( ( 1 - 17 p^{4} T + 20421 p^{3} T^{2} - 73757 p^{9} T^{3} + 13277881 p^{10} T^{4} - 73757 p^{25} T^{5} + 20421 p^{35} T^{6} - 17 p^{52} T^{7} + p^{64} T^{8} )^{2} \) |
| 5 | \( 1 - 128266937608 p T^{2} + \)\(16\!\cdots\!48\)\( p^{3} T^{4} - \)\(55\!\cdots\!64\)\( p^{7} T^{6} + \)\(37\!\cdots\!18\)\( p^{9} T^{8} - \)\(55\!\cdots\!64\)\( p^{39} T^{10} + \)\(16\!\cdots\!48\)\( p^{67} T^{12} - 128266937608 p^{97} T^{14} + p^{128} T^{16} \) | |
| 11 | \( ( 1 + 215199352 T + 6645512710994676 p T^{2} - \)\(11\!\cdots\!08\)\( p^{2} T^{3} + \)\(84\!\cdots\!94\)\( p^{3} T^{4} - \)\(11\!\cdots\!08\)\( p^{18} T^{5} + 6645512710994676 p^{33} T^{6} + 215199352 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 13 | \( 1 - 240153872372001608 p T^{2} + \)\(22\!\cdots\!32\)\( p^{3} T^{4} - \)\(14\!\cdots\!32\)\( p^{5} T^{6} + \)\(64\!\cdots\!90\)\( p^{7} T^{8} - \)\(14\!\cdots\!32\)\( p^{37} T^{10} + \)\(22\!\cdots\!32\)\( p^{67} T^{12} - 240153872372001608 p^{97} T^{14} + p^{128} T^{16} \) | |
| 17 | \( 1 - 34719280924735282184 T^{2} + \)\(31\!\cdots\!64\)\( T^{4} + \)\(28\!\cdots\!24\)\( T^{6} + \)\(90\!\cdots\!50\)\( T^{8} + \)\(28\!\cdots\!24\)\( p^{32} T^{10} + \)\(31\!\cdots\!64\)\( p^{64} T^{12} - 34719280924735282184 p^{96} T^{14} + p^{128} T^{16} \) | |
| 19 | \( 1 - \)\(24\!\cdots\!08\)\( T^{2} + \)\(75\!\cdots\!68\)\( T^{4} - \)\(27\!\cdots\!96\)\( T^{6} + \)\(13\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!96\)\( p^{32} T^{10} + \)\(75\!\cdots\!68\)\( p^{64} T^{12} - \)\(24\!\cdots\!08\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 23 | \( ( 1 + 44882541208 T + \)\(22\!\cdots\!08\)\( T^{2} + \)\(81\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!94\)\( T^{4} + \)\(81\!\cdots\!96\)\( p^{16} T^{5} + \)\(22\!\cdots\!08\)\( p^{32} T^{6} + 44882541208 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 29 | \( ( 1 - 11218795832 T + \)\(58\!\cdots\!36\)\( T^{2} - \)\(10\!\cdots\!92\)\( p T^{3} + \)\(16\!\cdots\!86\)\( T^{4} - \)\(10\!\cdots\!92\)\( p^{17} T^{5} + \)\(58\!\cdots\!36\)\( p^{32} T^{6} - 11218795832 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 31 | \( 1 - \)\(23\!\cdots\!08\)\( T^{2} + \)\(40\!\cdots\!68\)\( T^{4} - \)\(43\!\cdots\!96\)\( T^{6} + \)\(38\!\cdots\!70\)\( p^{2} T^{8} - \)\(43\!\cdots\!96\)\( p^{32} T^{10} + \)\(40\!\cdots\!68\)\( p^{64} T^{12} - \)\(23\!\cdots\!08\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 37 | \( ( 1 - 2868933267208 T + \)\(36\!\cdots\!28\)\( T^{2} - \)\(92\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!94\)\( T^{4} - \)\(92\!\cdots\!36\)\( p^{16} T^{5} + \)\(36\!\cdots\!28\)\( p^{32} T^{6} - 2868933267208 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 41 | \( 1 - \)\(37\!\cdots\!68\)\( T^{2} + \)\(66\!\cdots\!08\)\( T^{4} - \)\(74\!\cdots\!76\)\( T^{6} + \)\(57\!\cdots\!70\)\( T^{8} - \)\(74\!\cdots\!76\)\( p^{32} T^{10} + \)\(66\!\cdots\!08\)\( p^{64} T^{12} - \)\(37\!\cdots\!68\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 43 | \( ( 1 + 1988476055432 T + \)\(27\!\cdots\!28\)\( T^{2} + \)\(60\!\cdots\!64\)\( T^{3} + \)\(50\!\cdots\!54\)\( T^{4} + \)\(60\!\cdots\!64\)\( p^{16} T^{5} + \)\(27\!\cdots\!28\)\( p^{32} T^{6} + 1988476055432 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 47 | \( 1 - \)\(32\!\cdots\!44\)\( T^{2} + \)\(50\!\cdots\!84\)\( T^{4} - \)\(47\!\cdots\!96\)\( T^{6} + \)\(14\!\cdots\!50\)\( p^{2} T^{8} - \)\(47\!\cdots\!96\)\( p^{32} T^{10} + \)\(50\!\cdots\!84\)\( p^{64} T^{12} - \)\(32\!\cdots\!44\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 53 | \( ( 1 - 54339920753912 T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(61\!\cdots\!08\)\( T^{3} + \)\(51\!\cdots\!18\)\( T^{4} - \)\(61\!\cdots\!08\)\( p^{16} T^{5} + \)\(10\!\cdots\!60\)\( p^{32} T^{6} - 54339920753912 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(10\!\cdots\!68\)\( T^{2} + \)\(62\!\cdots\!08\)\( T^{4} - \)\(22\!\cdots\!76\)\( T^{6} + \)\(58\!\cdots\!70\)\( T^{8} - \)\(22\!\cdots\!76\)\( p^{32} T^{10} + \)\(62\!\cdots\!08\)\( p^{64} T^{12} - \)\(10\!\cdots\!68\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 61 | \( 1 - \)\(53\!\cdots\!28\)\( T^{2} + \)\(66\!\cdots\!28\)\( T^{4} - \)\(36\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(36\!\cdots\!36\)\( p^{32} T^{10} + \)\(66\!\cdots\!28\)\( p^{64} T^{12} - \)\(53\!\cdots\!28\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 67 | \( ( 1 + 361060032821512 T + \)\(55\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + \)\(17\!\cdots\!64\)\( p^{16} T^{5} + \)\(55\!\cdots\!88\)\( p^{32} T^{6} + 361060032821512 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 71 | \( ( 1 - 144547623334568 T + \)\(13\!\cdots\!96\)\( T^{2} - \)\(44\!\cdots\!88\)\( T^{3} + \)\(72\!\cdots\!34\)\( T^{4} - \)\(44\!\cdots\!88\)\( p^{16} T^{5} + \)\(13\!\cdots\!96\)\( p^{32} T^{6} - 144547623334568 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 73 | \( 1 - \)\(23\!\cdots\!84\)\( T^{2} + \)\(31\!\cdots\!64\)\( T^{4} - \)\(29\!\cdots\!36\)\( T^{6} + \)\(20\!\cdots\!30\)\( T^{8} - \)\(29\!\cdots\!36\)\( p^{32} T^{10} + \)\(31\!\cdots\!64\)\( p^{64} T^{12} - \)\(23\!\cdots\!84\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 79 | \( ( 1 - 2916388433281688 T + \)\(66\!\cdots\!56\)\( T^{2} - \)\(11\!\cdots\!92\)\( T^{3} + \)\(21\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!92\)\( p^{16} T^{5} + \)\(66\!\cdots\!56\)\( p^{32} T^{6} - 2916388433281688 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(24\!\cdots\!84\)\( T^{2} + \)\(29\!\cdots\!64\)\( T^{4} - \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!50\)\( T^{8} - \)\(24\!\cdots\!76\)\( p^{32} T^{10} + \)\(29\!\cdots\!64\)\( p^{64} T^{12} - \)\(24\!\cdots\!84\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 89 | \( 1 - \)\(42\!\cdots\!28\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{4} - \)\(23\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!70\)\( T^{8} - \)\(23\!\cdots\!36\)\( p^{32} T^{10} + \)\(12\!\cdots\!28\)\( p^{64} T^{12} - \)\(42\!\cdots\!28\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 97 | \( 1 - \)\(34\!\cdots\!44\)\( T^{2} + \)\(57\!\cdots\!84\)\( T^{4} - \)\(60\!\cdots\!96\)\( T^{6} + \)\(44\!\cdots\!50\)\( T^{8} - \)\(60\!\cdots\!96\)\( p^{32} T^{10} + \)\(57\!\cdots\!84\)\( p^{64} T^{12} - \)\(34\!\cdots\!44\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.93917112773409908798101270236, −3.91704900038615960798617220098, −3.91038683664612474136956708718, −3.86637974069746119292030022827, −3.81015047484428867112472863071, −3.32590682367540110951201334187, −3.15522008299843173680594825844, −2.89078122965384934770826035533, −2.84258229956215859559559307082, −2.75129241757026826799007387620, −2.62677656044812775610496722426, −2.60895151646581411902151444974, −2.25664333549556389685401697897, −2.10009445932866879270479026167, −1.68581073332447912399206105983, −1.56348436323714255429131478051, −1.38456515674711214592221405328, −1.23521096388826658563477191724, −1.08388166200213354976266829927, −0.77973206489182396208782882203, −0.75127569303863225850114154205, −0.48436009077929397464850344782, −0.29406543592084604025834504043, −0.29371643452772649159283413076, −0.18915634748325211098153833820, 0.18915634748325211098153833820, 0.29371643452772649159283413076, 0.29406543592084604025834504043, 0.48436009077929397464850344782, 0.75127569303863225850114154205, 0.77973206489182396208782882203, 1.08388166200213354976266829927, 1.23521096388826658563477191724, 1.38456515674711214592221405328, 1.56348436323714255429131478051, 1.68581073332447912399206105983, 2.10009445932866879270479026167, 2.25664333549556389685401697897, 2.60895151646581411902151444974, 2.62677656044812775610496722426, 2.75129241757026826799007387620, 2.84258229956215859559559307082, 2.89078122965384934770826035533, 3.15522008299843173680594825844, 3.32590682367540110951201334187, 3.81015047484428867112472863071, 3.86637974069746119292030022827, 3.91038683664612474136956708718, 3.91704900038615960798617220098, 3.93917112773409908798101270236
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.