Dirichlet series
| L(s) = 1 | − 9.87e5·3-s + 3.33e11·7-s + 2.65e11·9-s − 1.13e14·13-s + 2.21e16·19-s − 3.28e17·21-s + 3.94e18·25-s − 3.72e18·27-s + 3.55e19·31-s + 4.82e20·37-s + 1.11e20·39-s − 3.04e20·43-s + 5.99e21·49-s − 2.18e22·57-s + 7.91e22·61-s + 8.85e22·63-s + 7.80e23·67-s − 3.41e24·73-s − 3.90e24·75-s − 7.99e24·79-s + 2.84e24·81-s − 3.77e25·91-s − 3.51e25·93-s − 1.31e26·97-s − 2.78e26·103-s + 7.58e25·109-s − 4.76e26·111-s + ⋯ |
| L(s) = 1 | − 0.619·3-s + 3.43·7-s + 0.104·9-s − 0.374·13-s + 0.525·19-s − 2.12·21-s + 2.65·25-s − 0.918·27-s + 1.45·31-s + 1.98·37-s + 0.231·39-s − 0.177·43-s + 0.638·49-s − 0.325·57-s + 0.489·61-s + 0.359·63-s + 1.42·67-s − 2.04·73-s − 1.64·75-s − 1.71·79-s + 0.440·81-s − 1.28·91-s − 0.902·93-s − 1.95·97-s − 1.89·103-s + 0.247·109-s − 1.22·111-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(27-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+13)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.19048\times 10^{18}\) |
| Root analytic conductor: | \(14.3380\) |
| Motivic weight: | \(26\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} ,\ ( \ : [13]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{27}{2})\) | \(\approx\) | \(0.07586511869\) |
| \(L(\frac12)\) | \(\approx\) | \(0.07586511869\) |
| \(L(14)\) | 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 | \( 1 + 329240 p T + 2921287508 p^{5} T^{2} + 7828870328120 p^{12} T^{3} + 454460465094722 p^{21} T^{4} + 7828870328120 p^{38} T^{5} + 2921287508 p^{57} T^{6} + 329240 p^{79} T^{7} + p^{104} T^{8} \) | |
| good | 5 | \( 1 - 789970737483985576 p T^{2} + \)\(39\!\cdots\!72\)\( p^{5} T^{4} - \)\(27\!\cdots\!76\)\( p^{10} T^{6} + \)\(14\!\cdots\!22\)\( p^{15} T^{8} - \)\(27\!\cdots\!76\)\( p^{62} T^{10} + \)\(39\!\cdots\!72\)\( p^{109} T^{12} - 789970737483985576 p^{157} T^{14} + p^{208} T^{16} \) |
| 7 | \( ( 1 - 23790801640 p T + \)\(55\!\cdots\!44\)\( p T^{2} - \)\(11\!\cdots\!80\)\( p^{3} T^{3} + \)\(43\!\cdots\!82\)\( p^{6} T^{4} - \)\(11\!\cdots\!80\)\( p^{29} T^{5} + \)\(55\!\cdots\!44\)\( p^{53} T^{6} - 23790801640 p^{79} T^{7} + p^{104} T^{8} )^{2} \) | |
| 11 | \( 1 - \)\(35\!\cdots\!28\)\( p T^{2} + \)\(62\!\cdots\!48\)\( p^{2} T^{4} - \)\(54\!\cdots\!76\)\( p^{6} T^{6} + \)\(45\!\cdots\!70\)\( p^{9} T^{8} - \)\(54\!\cdots\!76\)\( p^{58} T^{10} + \)\(62\!\cdots\!48\)\( p^{106} T^{12} - \)\(35\!\cdots\!28\)\( p^{157} T^{14} + p^{208} T^{16} \) | |
| 13 | \( ( 1 + 4360783516600 p T + \)\(44\!\cdots\!16\)\( p T^{2} + \)\(10\!\cdots\!00\)\( p^{2} T^{3} + \)\(25\!\cdots\!62\)\( p^{2} T^{4} + \)\(10\!\cdots\!00\)\( p^{28} T^{5} + \)\(44\!\cdots\!16\)\( p^{53} T^{6} + 4360783516600 p^{79} T^{7} + p^{104} T^{8} )^{2} \) | |
| 17 | \( 1 - \)\(11\!\cdots\!72\)\( T^{2} + \)\(76\!\cdots\!64\)\( p T^{4} - \)\(39\!\cdots\!08\)\( p^{3} T^{6} + \)\(13\!\cdots\!10\)\( p^{5} T^{8} - \)\(39\!\cdots\!08\)\( p^{55} T^{10} + \)\(76\!\cdots\!64\)\( p^{105} T^{12} - \)\(11\!\cdots\!72\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 19 | \( ( 1 - 11053030303349176 T + \)\(64\!\cdots\!60\)\( p T^{2} - \)\(11\!\cdots\!64\)\( p^{2} T^{3} + \)\(71\!\cdots\!06\)\( p^{3} T^{4} - \)\(11\!\cdots\!64\)\( p^{28} T^{5} + \)\(64\!\cdots\!60\)\( p^{53} T^{6} - 11053030303349176 p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 23 | \( 1 - \)\(39\!\cdots\!04\)\( p T^{2} + \)\(32\!\cdots\!24\)\( p^{3} T^{4} - \)\(17\!\cdots\!48\)\( p^{5} T^{6} + \)\(78\!\cdots\!10\)\( p^{7} T^{8} - \)\(17\!\cdots\!48\)\( p^{57} T^{10} + \)\(32\!\cdots\!24\)\( p^{107} T^{12} - \)\(39\!\cdots\!04\)\( p^{157} T^{14} + p^{208} T^{16} \) | |
| 29 | \( 1 - \)\(40\!\cdots\!48\)\( T^{2} + \)\(10\!\cdots\!28\)\( T^{4} - \)\(17\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!70\)\( T^{8} - \)\(17\!\cdots\!16\)\( p^{52} T^{10} + \)\(10\!\cdots\!28\)\( p^{104} T^{12} - \)\(40\!\cdots\!48\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 31 | \( ( 1 - 17777908834640091544 T + \)\(20\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!94\)\( T^{4} - \)\(22\!\cdots\!16\)\( p^{26} T^{5} + \)\(20\!\cdots\!00\)\( p^{52} T^{6} - 17777908834640091544 p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 37 | \( ( 1 - \)\(24\!\cdots\!60\)\( T + \)\(20\!\cdots\!84\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!26\)\( T^{4} - \)\(33\!\cdots\!60\)\( p^{26} T^{5} + \)\(20\!\cdots\!84\)\( p^{52} T^{6} - \)\(24\!\cdots\!60\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 41 | \( 1 - \)\(43\!\cdots\!08\)\( T^{2} + \)\(98\!\cdots\!48\)\( T^{4} - \)\(14\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!70\)\( T^{8} - \)\(14\!\cdots\!76\)\( p^{52} T^{10} + \)\(98\!\cdots\!48\)\( p^{104} T^{12} - \)\(43\!\cdots\!08\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 43 | \( ( 1 + \)\(15\!\cdots\!20\)\( T + \)\(11\!\cdots\!28\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(48\!\cdots\!98\)\( T^{4} + \)\(12\!\cdots\!60\)\( p^{26} T^{5} + \)\(11\!\cdots\!28\)\( p^{52} T^{6} + \)\(15\!\cdots\!20\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 47 | \( 1 - \)\(12\!\cdots\!12\)\( T^{2} + \)\(67\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{6} + \)\(75\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!84\)\( p^{52} T^{10} + \)\(67\!\cdots\!68\)\( p^{104} T^{12} - \)\(12\!\cdots\!12\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 53 | \( 1 - \)\(20\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{6} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!84\)\( p^{52} T^{10} + \)\(26\!\cdots\!68\)\( p^{104} T^{12} - \)\(20\!\cdots\!12\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 59 | \( 1 - \)\(33\!\cdots\!48\)\( T^{2} + \)\(39\!\cdots\!88\)\( T^{4} - \)\(38\!\cdots\!76\)\( T^{6} - \)\(26\!\cdots\!30\)\( T^{8} - \)\(38\!\cdots\!76\)\( p^{52} T^{10} + \)\(39\!\cdots\!88\)\( p^{104} T^{12} - \)\(33\!\cdots\!48\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 61 | \( ( 1 - \)\(39\!\cdots\!16\)\( T + \)\(26\!\cdots\!40\)\( T^{2} + \)\(92\!\cdots\!16\)\( T^{3} + \)\(88\!\cdots\!94\)\( T^{4} + \)\(92\!\cdots\!16\)\( p^{26} T^{5} + \)\(26\!\cdots\!40\)\( p^{52} T^{6} - \)\(39\!\cdots\!16\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 67 | \( ( 1 - \)\(39\!\cdots\!00\)\( T + \)\(56\!\cdots\!08\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(11\!\cdots\!00\)\( p^{26} T^{5} + \)\(56\!\cdots\!08\)\( p^{52} T^{6} - \)\(39\!\cdots\!00\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 71 | \( 1 - \)\(92\!\cdots\!88\)\( T^{2} + \)\(45\!\cdots\!68\)\( T^{4} - \)\(55\!\cdots\!16\)\( T^{6} + \)\(98\!\cdots\!70\)\( T^{8} - \)\(55\!\cdots\!16\)\( p^{52} T^{10} + \)\(45\!\cdots\!68\)\( p^{104} T^{12} - \)\(92\!\cdots\!88\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 73 | \( ( 1 + \)\(17\!\cdots\!20\)\( T + \)\(10\!\cdots\!84\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!06\)\( T^{4} + \)\(13\!\cdots\!20\)\( p^{26} T^{5} + \)\(10\!\cdots\!84\)\( p^{52} T^{6} + \)\(17\!\cdots\!20\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 79 | \( ( 1 + \)\(39\!\cdots\!32\)\( T + \)\(86\!\cdots\!68\)\( T^{2} + \)\(25\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!64\)\( p^{26} T^{5} + \)\(86\!\cdots\!68\)\( p^{52} T^{6} + \)\(39\!\cdots\!32\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
| 83 | \( 1 - \)\(38\!\cdots\!72\)\( T^{2} + \)\(75\!\cdots\!88\)\( T^{4} - \)\(97\!\cdots\!04\)\( T^{6} + \)\(90\!\cdots\!70\)\( T^{8} - \)\(97\!\cdots\!04\)\( p^{52} T^{10} + \)\(75\!\cdots\!88\)\( p^{104} T^{12} - \)\(38\!\cdots\!72\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 89 | \( 1 - \)\(14\!\cdots\!68\)\( T^{2} + \)\(10\!\cdots\!68\)\( T^{4} - \)\(65\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!70\)\( T^{8} - \)\(65\!\cdots\!36\)\( p^{52} T^{10} + \)\(10\!\cdots\!68\)\( p^{104} T^{12} - \)\(14\!\cdots\!68\)\( p^{156} T^{14} + p^{208} T^{16} \) | |
| 97 | \( ( 1 + \)\(65\!\cdots\!80\)\( T + \)\(11\!\cdots\!08\)\( T^{2} + \)\(39\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!98\)\( T^{4} + \)\(39\!\cdots\!40\)\( p^{26} T^{5} + \)\(11\!\cdots\!08\)\( p^{52} T^{6} + \)\(65\!\cdots\!80\)\( p^{78} T^{7} + p^{104} T^{8} )^{2} \) | |
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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
−3.77154678093346354469663023140, −3.42309423260749655832328088277, −3.16391789523270653234017937488, −3.03287165533654924106712030841, −2.92034613835418903664734177458, −2.86566126964876020786435065474, −2.86270995696359659489464907313, −2.55476402398015356243109324516, −2.48145042734591222290790499798, −2.11254281544949287414893262366, −2.06008196414506298022412129656, −1.91198138909031253220094409544, −1.84881604188086598761502348771, −1.67515013614915095036430407270, −1.48875736288538871039488678736, −1.41002171212232436976679228082, −1.16080016765702905080467437156, −1.12596681398679846166762688771, −0.929968257008556921537639115408, −0.905450350359382711560086084299, −0.819080973698976533402627472782, −0.808370949622840602263609072936, −0.13921351205664596658697075606, −0.088550039689077239895107330627, −0.05484095982673006938129773731, 0.05484095982673006938129773731, 0.088550039689077239895107330627, 0.13921351205664596658697075606, 0.808370949622840602263609072936, 0.819080973698976533402627472782, 0.905450350359382711560086084299, 0.929968257008556921537639115408, 1.12596681398679846166762688771, 1.16080016765702905080467437156, 1.41002171212232436976679228082, 1.48875736288538871039488678736, 1.67515013614915095036430407270, 1.84881604188086598761502348771, 1.91198138909031253220094409544, 2.06008196414506298022412129656, 2.11254281544949287414893262366, 2.48145042734591222290790499798, 2.55476402398015356243109324516, 2.86270995696359659489464907313, 2.86566126964876020786435065474, 2.92034613835418903664734177458, 3.03287165533654924106712030841, 3.16391789523270653234017937488, 3.42309423260749655832328088277, 3.77154678093346354469663023140
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.