Dirichlet series
| L(s) = 1 | − 1.12e6·5-s + 7.76e7·9-s + 1.76e7·13-s − 7.98e9·17-s + 9.84e10·25-s − 1.91e11·29-s + 2.82e12·37-s + 7.72e12·41-s − 8.70e13·45-s + 1.50e14·49-s + 1.85e14·53-s + 1.58e15·61-s − 1.97e13·65-s + 4.73e15·73-s + 4.18e15·81-s + 8.95e15·85-s − 4.26e15·89-s − 1.60e16·97-s − 2.83e16·101-s − 5.97e16·109-s − 1.25e16·113-s + 1.36e15·117-s + 1.05e17·121-s + 3.96e17·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.87·5-s + 1.80·9-s + 0.0216·13-s − 1.14·17-s + 0.645·25-s − 0.383·29-s + 0.803·37-s + 0.967·41-s − 5.17·45-s + 4.54·49-s + 2.98·53-s + 8.28·61-s − 0.0620·65-s + 5.87·73-s + 2.25·81-s + 3.28·85-s − 1.08·89-s − 2.04·97-s − 2.61·101-s − 3.00·109-s − 0.472·113-s + 0.0389·117-s + 2.29·121-s + 6.65·125-s + 1.10·145-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(17-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+8)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{40}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.29996\times 10^{13}\) |
| Root analytic conductor: | \(7.20720\) |
| Motivic weight: | \(16\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{40} ,\ ( \ : [8]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{17}{2})\) | \(\approx\) | \(10.91331560\) |
| \(L(\frac12)\) | \(\approx\) | \(10.91331560\) |
| \(L(9)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 77615048 T^{2} + 68131462831508 p^{3} T^{4} - 83596113906978867512 p^{6} T^{6} + \)\(93\!\cdots\!66\)\( p^{14} T^{8} - 83596113906978867512 p^{38} T^{10} + 68131462831508 p^{67} T^{12} - 77615048 p^{96} T^{14} + p^{128} T^{16} \) |
| 5 | \( ( 1 + 560728 T + 84477175724 p T^{2} + 1277425335245192 p^{3} T^{3} + \)\(13\!\cdots\!18\)\( p^{4} T^{4} + 1277425335245192 p^{19} T^{5} + 84477175724 p^{33} T^{6} + 560728 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 7 | \( 1 - 150920818940680 T^{2} + \)\(23\!\cdots\!40\)\( p^{2} T^{4} - \)\(48\!\cdots\!76\)\( p^{6} T^{6} + \)\(75\!\cdots\!30\)\( p^{10} T^{8} - \)\(48\!\cdots\!76\)\( p^{38} T^{10} + \)\(23\!\cdots\!40\)\( p^{66} T^{12} - 150920818940680 p^{96} T^{14} + p^{128} T^{16} \) | |
| 11 | \( 1 - 105408625380378568 T^{2} + \)\(42\!\cdots\!12\)\( p^{2} T^{4} - \)\(99\!\cdots\!72\)\( p^{4} T^{6} + \)\(25\!\cdots\!54\)\( p^{6} T^{8} - \)\(99\!\cdots\!72\)\( p^{36} T^{10} + \)\(42\!\cdots\!12\)\( p^{66} T^{12} - 105408625380378568 p^{96} T^{14} + p^{128} T^{16} \) | |
| 13 | \( ( 1 - 8813608 T + 94188628794062284 p T^{2} + \)\(53\!\cdots\!84\)\( p^{2} T^{3} + \)\(26\!\cdots\!50\)\( p^{3} T^{4} + \)\(53\!\cdots\!84\)\( p^{18} T^{5} + 94188628794062284 p^{33} T^{6} - 8813608 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 17 | \( ( 1 + 3991515960 T + \)\(16\!\cdots\!88\)\( T^{2} + \)\(55\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!94\)\( T^{4} + \)\(55\!\cdots\!80\)\( p^{16} T^{5} + \)\(16\!\cdots\!88\)\( p^{32} T^{6} + 3991515960 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 19 | \( 1 - \)\(13\!\cdots\!52\)\( T^{2} + \)\(92\!\cdots\!08\)\( T^{4} - \)\(42\!\cdots\!04\)\( T^{6} + \)\(39\!\cdots\!70\)\( p^{2} T^{8} - \)\(42\!\cdots\!04\)\( p^{32} T^{10} + \)\(92\!\cdots\!08\)\( p^{64} T^{12} - \)\(13\!\cdots\!52\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 23 | \( 1 + \)\(41\!\cdots\!68\)\( T^{2} + \)\(38\!\cdots\!72\)\( p T^{4} + \)\(13\!\cdots\!52\)\( T^{6} + \)\(40\!\cdots\!06\)\( T^{8} + \)\(13\!\cdots\!52\)\( p^{32} T^{10} + \)\(38\!\cdots\!72\)\( p^{65} T^{12} + \)\(41\!\cdots\!68\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 29 | \( ( 1 + 95985037400 T + \)\(83\!\cdots\!32\)\( T^{2} + \)\(53\!\cdots\!64\)\( T^{3} + \)\(29\!\cdots\!62\)\( T^{4} + \)\(53\!\cdots\!64\)\( p^{16} T^{5} + \)\(83\!\cdots\!32\)\( p^{32} T^{6} + 95985037400 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 31 | \( 1 - \)\(56\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(21\!\cdots\!48\)\( p^{2} T^{6} + \)\(19\!\cdots\!66\)\( p^{4} T^{8} - \)\(21\!\cdots\!48\)\( p^{34} T^{10} + \)\(14\!\cdots\!56\)\( p^{64} T^{12} - \)\(56\!\cdots\!32\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 37 | \( ( 1 - 38160192264 p T + \)\(41\!\cdots\!12\)\( T^{2} - \)\(56\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!02\)\( T^{4} - \)\(56\!\cdots\!00\)\( p^{16} T^{5} + \)\(41\!\cdots\!12\)\( p^{32} T^{6} - 38160192264 p^{49} T^{7} + p^{64} T^{8} )^{2} \) | |
| 41 | \( ( 1 - 3864560283016 T + \)\(25\!\cdots\!04\)\( T^{2} - \)\(64\!\cdots\!28\)\( T^{3} + \)\(45\!\cdots\!66\)\( T^{4} - \)\(64\!\cdots\!28\)\( p^{16} T^{5} + \)\(25\!\cdots\!04\)\( p^{32} T^{6} - 3864560283016 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 43 | \( 1 - \)\(56\!\cdots\!24\)\( T^{2} + \)\(12\!\cdots\!68\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{6} + \)\(12\!\cdots\!22\)\( T^{8} - \)\(13\!\cdots\!56\)\( p^{32} T^{10} + \)\(12\!\cdots\!68\)\( p^{64} T^{12} - \)\(56\!\cdots\!24\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 47 | \( 1 - \)\(14\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{4} - \)\(85\!\cdots\!84\)\( T^{6} + \)\(49\!\cdots\!70\)\( T^{8} - \)\(85\!\cdots\!84\)\( p^{32} T^{10} + \)\(13\!\cdots\!68\)\( p^{64} T^{12} - \)\(14\!\cdots\!12\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 53 | \( ( 1 - 92881681952040 T + \)\(55\!\cdots\!96\)\( T^{2} + \)\(15\!\cdots\!72\)\( T^{3} - \)\(16\!\cdots\!70\)\( T^{4} + \)\(15\!\cdots\!72\)\( p^{16} T^{5} + \)\(55\!\cdots\!96\)\( p^{32} T^{6} - 92881681952040 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 59 | \( 1 - \)\(11\!\cdots\!40\)\( T^{2} + \)\(64\!\cdots\!60\)\( T^{4} - \)\(23\!\cdots\!84\)\( T^{6} + \)\(60\!\cdots\!90\)\( T^{8} - \)\(23\!\cdots\!84\)\( p^{32} T^{10} + \)\(64\!\cdots\!60\)\( p^{64} T^{12} - \)\(11\!\cdots\!40\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 61 | \( ( 1 - 794147593645096 T + \)\(35\!\cdots\!12\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!80\)\( p^{16} T^{5} + \)\(35\!\cdots\!12\)\( p^{32} T^{6} - 794147593645096 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 67 | \( 1 - \)\(65\!\cdots\!28\)\( T^{2} + \)\(20\!\cdots\!92\)\( T^{4} - \)\(40\!\cdots\!12\)\( T^{6} + \)\(68\!\cdots\!74\)\( T^{8} - \)\(40\!\cdots\!12\)\( p^{32} T^{10} + \)\(20\!\cdots\!92\)\( p^{64} T^{12} - \)\(65\!\cdots\!28\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 71 | \( 1 - \)\(21\!\cdots\!80\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{4} - \)\(13\!\cdots\!84\)\( T^{6} + \)\(65\!\cdots\!30\)\( T^{8} - \)\(13\!\cdots\!84\)\( p^{32} T^{10} + \)\(21\!\cdots\!60\)\( p^{64} T^{12} - \)\(21\!\cdots\!80\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 73 | \( ( 1 - 2367842322225864 T + \)\(36\!\cdots\!32\)\( T^{2} - \)\(40\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!86\)\( T^{4} - \)\(40\!\cdots\!60\)\( p^{16} T^{5} + \)\(36\!\cdots\!32\)\( p^{32} T^{6} - 2367842322225864 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 79 | \( 1 - \)\(88\!\cdots\!48\)\( T^{2} + \)\(50\!\cdots\!96\)\( T^{4} - \)\(18\!\cdots\!68\)\( T^{6} + \)\(79\!\cdots\!54\)\( p^{2} T^{8} - \)\(18\!\cdots\!68\)\( p^{32} T^{10} + \)\(50\!\cdots\!96\)\( p^{64} T^{12} - \)\(88\!\cdots\!48\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 83 | \( 1 - \)\(17\!\cdots\!56\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{6} + \)\(19\!\cdots\!86\)\( T^{8} - \)\(50\!\cdots\!00\)\( p^{32} T^{10} + \)\(12\!\cdots\!12\)\( p^{64} T^{12} - \)\(17\!\cdots\!56\)\( p^{96} T^{14} + p^{128} T^{16} \) | |
| 89 | \( ( 1 + 2133980684705592 T + \)\(56\!\cdots\!04\)\( T^{2} + \)\(97\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!46\)\( T^{4} + \)\(97\!\cdots\!36\)\( p^{16} T^{5} + \)\(56\!\cdots\!04\)\( p^{32} T^{6} + 2133980684705592 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
| 97 | \( ( 1 + 8011487951838776 T + \)\(13\!\cdots\!44\)\( T^{2} + \)\(88\!\cdots\!12\)\( p T^{3} + \)\(48\!\cdots\!50\)\( T^{4} + \)\(88\!\cdots\!12\)\( p^{17} T^{5} + \)\(13\!\cdots\!44\)\( p^{32} T^{6} + 8011487951838776 p^{48} T^{7} + p^{64} T^{8} )^{2} \) | |
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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
−4.64493108399187992628172131545, −4.19696398252359586035983073338, −4.12676882257211828013036914109, −4.07752096843611863361843207020, −4.02061516916791653684194180678, −4.02053999348713846538164938406, −3.82590330546365601601661761893, −3.62195419964738904947263366813, −3.35869895965424607285416877588, −3.32533721233090034429773653616, −2.66017766464051153712470760407, −2.52625297099219783414652978835, −2.50774697072005212172303398595, −2.18947492984106266787952316849, −2.09009841627377888460967752653, −2.05747153682340354106658558534, −1.88902681391001037966180775261, −1.26056761205007555103905562965, −1.15219327190667999203525485587, −0.973369387744232351908429158703, −0.824306058211520933095825555846, −0.64302509466326376259400254133, −0.46502023751953663884970447002, −0.40618720494930115426948904691, −0.26086319111590707743761718528, 0.26086319111590707743761718528, 0.40618720494930115426948904691, 0.46502023751953663884970447002, 0.64302509466326376259400254133, 0.824306058211520933095825555846, 0.973369387744232351908429158703, 1.15219327190667999203525485587, 1.26056761205007555103905562965, 1.88902681391001037966180775261, 2.05747153682340354106658558534, 2.09009841627377888460967752653, 2.18947492984106266787952316849, 2.50774697072005212172303398595, 2.52625297099219783414652978835, 2.66017766464051153712470760407, 3.32533721233090034429773653616, 3.35869895965424607285416877588, 3.62195419964738904947263366813, 3.82590330546365601601661761893, 4.02053999348713846538164938406, 4.02061516916791653684194180678, 4.07752096843611863361843207020, 4.12676882257211828013036914109, 4.19696398252359586035983073338, 4.64493108399187992628172131545
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.