Dirichlet series
| L(s) = 1 | − 456·7-s − 7.50e4·13-s + 1.32e5·19-s + 9.68e5·25-s + 1.30e6·31-s − 2.55e6·37-s + 2.95e6·43-s − 1.37e7·49-s + 2.06e7·61-s − 5.91e7·67-s − 9.87e7·73-s + 1.07e8·79-s + 3.42e7·91-s − 1.03e8·97-s + 6.31e6·103-s + 2.45e8·109-s + 1.09e9·121-s + ⋯ |
| L(s) = 1 | − 0.189·7-s − 2.62·13-s + 1.01·19-s + 2.47·25-s + 1.40·31-s − 1.36·37-s + 0.865·43-s − 2.38·49-s + 1.49·61-s − 2.93·67-s − 3.47·73-s + 2.75·79-s + 0.498·91-s − 1.17·97-s + 0.0560·103-s + 1.74·109-s + 5.12·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.20143\times 10^{17}\) |
| Root analytic conductor: | \(13.2660\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{24} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(27.05899680\) |
| \(L(\frac12)\) | \(\approx\) | \(27.05899680\) |
| \(L(5)\) | 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 | \( 1 \) | |
| good | 5 | \( 1 - 968616 T^{2} + 496942836316 T^{4} - 5164354233440856 p^{2} T^{6} + 58700931098313842214 p^{4} T^{8} - 5164354233440856 p^{18} T^{10} + 496942836316 p^{32} T^{12} - 968616 p^{48} T^{14} + p^{64} T^{16} \) |
| 7 | \( ( 1 + 228 T + 141898 p^{2} T^{2} + 22666320 p^{2} T^{3} + 96671289093 p^{3} T^{4} + 22666320 p^{10} T^{5} + 141898 p^{18} T^{6} + 228 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 11 | \( 1 - 1098083496 T^{2} + 52243307117245076 p T^{4} - \)\(19\!\cdots\!48\)\( T^{6} + \)\(47\!\cdots\!30\)\( T^{8} - \)\(19\!\cdots\!48\)\( p^{16} T^{10} + 52243307117245076 p^{33} T^{12} - 1098083496 p^{48} T^{14} + p^{64} T^{16} \) | |
| 13 | \( ( 1 + 37508 T + 1620476170 T^{2} + 4005848565968 p T^{3} + 2180879276314990483 T^{4} + 4005848565968 p^{9} T^{5} + 1620476170 p^{16} T^{6} + 37508 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 17 | \( 1 - 34035069864 T^{2} + \)\(59\!\cdots\!68\)\( T^{4} - \)\(68\!\cdots\!92\)\( T^{6} + \)\(55\!\cdots\!94\)\( T^{8} - \)\(68\!\cdots\!92\)\( p^{16} T^{10} + \)\(59\!\cdots\!68\)\( p^{32} T^{12} - 34035069864 p^{48} T^{14} + p^{64} T^{16} \) | |
| 19 | \( ( 1 - 66268 T + 29150019946 T^{2} - 912723957742768 T^{3} + \)\(41\!\cdots\!39\)\( T^{4} - 912723957742768 p^{8} T^{5} + 29150019946 p^{16} T^{6} - 66268 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 23 | \( 1 + 85834704984 T^{2} + \)\(52\!\cdots\!16\)\( T^{4} - \)\(87\!\cdots\!28\)\( T^{6} - \)\(43\!\cdots\!10\)\( T^{8} - \)\(87\!\cdots\!28\)\( p^{16} T^{10} + \)\(52\!\cdots\!16\)\( p^{32} T^{12} + 85834704984 p^{48} T^{14} + p^{64} T^{16} \) | |
| 29 | \( 1 - 2889513884552 T^{2} + \)\(39\!\cdots\!00\)\( T^{4} - \)\(34\!\cdots\!76\)\( T^{6} + \)\(20\!\cdots\!78\)\( T^{8} - \)\(34\!\cdots\!76\)\( p^{16} T^{10} + \)\(39\!\cdots\!00\)\( p^{32} T^{12} - 2889513884552 p^{48} T^{14} + p^{64} T^{16} \) | |
| 31 | \( ( 1 - 650536 T + 2764567712860 T^{2} - 1666580922887428888 T^{3} + \)\(32\!\cdots\!98\)\( T^{4} - 1666580922887428888 p^{8} T^{5} + 2764567712860 p^{16} T^{6} - 650536 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 1279716 T + 7553234320042 T^{2} + 13987141328200758480 T^{3} + \)\(28\!\cdots\!15\)\( T^{4} + 13987141328200758480 p^{8} T^{5} + 7553234320042 p^{16} T^{6} + 1279716 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 41 | \( 1 - 47607926533256 T^{2} + \)\(11\!\cdots\!96\)\( T^{4} - \)\(15\!\cdots\!48\)\( T^{6} + \)\(15\!\cdots\!30\)\( T^{8} - \)\(15\!\cdots\!48\)\( p^{16} T^{10} + \)\(11\!\cdots\!96\)\( p^{32} T^{12} - 47607926533256 p^{48} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 - 1479272 T + 25544220734812 T^{2} - 50645098163676178136 T^{3} + \)\(39\!\cdots\!02\)\( T^{4} - 50645098163676178136 p^{8} T^{5} + 25544220734812 p^{16} T^{6} - 1479272 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 - 120469666303400 T^{2} + \)\(71\!\cdots\!56\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{6} + \)\(78\!\cdots\!66\)\( T^{8} - \)\(27\!\cdots\!00\)\( p^{16} T^{10} + \)\(71\!\cdots\!56\)\( p^{32} T^{12} - 120469666303400 p^{48} T^{14} + p^{64} T^{16} \) | |
| 53 | \( 1 - 231235989016968 T^{2} + \)\(24\!\cdots\!92\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{6} + \)\(98\!\cdots\!82\)\( T^{8} - \)\(16\!\cdots\!44\)\( p^{16} T^{10} + \)\(24\!\cdots\!92\)\( p^{32} T^{12} - 231235989016968 p^{48} T^{14} + p^{64} T^{16} \) | |
| 59 | \( 1 - 517964583880104 T^{2} + \)\(15\!\cdots\!20\)\( T^{4} - \)\(32\!\cdots\!40\)\( T^{6} + \)\(52\!\cdots\!66\)\( T^{8} - \)\(32\!\cdots\!40\)\( p^{16} T^{10} + \)\(15\!\cdots\!20\)\( p^{32} T^{12} - 517964583880104 p^{48} T^{14} + p^{64} T^{16} \) | |
| 61 | \( ( 1 - 10331836 T + 698853665923402 T^{2} - \)\(52\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!35\)\( T^{4} - \)\(52\!\cdots\!80\)\( p^{8} T^{5} + 698853665923402 p^{16} T^{6} - 10331836 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 67 | \( ( 1 + 29584196 T + 844213121517130 T^{2} + \)\(21\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!59\)\( T^{4} + \)\(21\!\cdots\!84\)\( p^{8} T^{5} + 844213121517130 p^{16} T^{6} + 29584196 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 71 | \( 1 - 2868447299584008 T^{2} + \)\(40\!\cdots\!12\)\( T^{4} - \)\(39\!\cdots\!84\)\( T^{6} + \)\(28\!\cdots\!02\)\( T^{8} - \)\(39\!\cdots\!84\)\( p^{16} T^{10} + \)\(40\!\cdots\!12\)\( p^{32} T^{12} - 2868447299584008 p^{48} T^{14} + p^{64} T^{16} \) | |
| 73 | \( ( 1 + 49378820 T + 1724028870691594 T^{2} + \)\(84\!\cdots\!84\)\( T^{3} + \)\(40\!\cdots\!39\)\( T^{4} + \)\(84\!\cdots\!84\)\( p^{8} T^{5} + 1724028870691594 p^{16} T^{6} + 49378820 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 79 | \( ( 1 - 53624444 T + 4725220190102410 T^{2} - \)\(22\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(22\!\cdots\!44\)\( p^{8} T^{5} + 4725220190102410 p^{16} T^{6} - 53624444 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 83 | \( 1 - 13242965355236232 T^{2} + \)\(84\!\cdots\!68\)\( T^{4} - \)\(33\!\cdots\!36\)\( T^{6} + \)\(91\!\cdots\!94\)\( T^{8} - \)\(33\!\cdots\!36\)\( p^{16} T^{10} + \)\(84\!\cdots\!68\)\( p^{32} T^{12} - 13242965355236232 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 - 19472990899110056 T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(86\!\cdots\!52\)\( T^{6} + \)\(35\!\cdots\!74\)\( T^{8} - \)\(86\!\cdots\!52\)\( p^{16} T^{10} + \)\(16\!\cdots\!00\)\( p^{32} T^{12} - 19472990899110056 p^{48} T^{14} + p^{64} T^{16} \) | |
| 97 | \( ( 1 + 51807652 T + 16344790227967594 T^{2} + \)\(18\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!83\)\( T^{4} + \)\(18\!\cdots\!36\)\( p^{8} T^{5} + 16344790227967594 p^{16} T^{6} + 51807652 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
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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
−3.59173199053653741745129210331, −3.19909678213323362108365243633, −3.02485739121450765925308870027, −3.01084289608797837782689905202, −2.97874275431222811251140038664, −2.92683364880975113733159737286, −2.91574612899314976555252970988, −2.88346498717756419953465572281, −2.56355783777222398001366205485, −2.20145900043412759520930840966, −2.17774188739654595733060950170, −1.84893780742691448368183324519, −1.82431275437480476857695987459, −1.72645350195011863935305548703, −1.68051463452563523618063487310, −1.64615339703364234342185691890, −1.27340998121651347104172286911, −1.17520367771101033795350199835, −0.73891912677531575771037053032, −0.66946989332501059826268560482, −0.62732980228317681556963413469, −0.55546021689333478391335349452, −0.47038774335821194750469098482, −0.45563408663391305600265980972, −0.16851911781016952325277452690, 0.16851911781016952325277452690, 0.45563408663391305600265980972, 0.47038774335821194750469098482, 0.55546021689333478391335349452, 0.62732980228317681556963413469, 0.66946989332501059826268560482, 0.73891912677531575771037053032, 1.17520367771101033795350199835, 1.27340998121651347104172286911, 1.64615339703364234342185691890, 1.68051463452563523618063487310, 1.72645350195011863935305548703, 1.82431275437480476857695987459, 1.84893780742691448368183324519, 2.17774188739654595733060950170, 2.20145900043412759520930840966, 2.56355783777222398001366205485, 2.88346498717756419953465572281, 2.91574612899314976555252970988, 2.92683364880975113733159737286, 2.97874275431222811251140038664, 3.01084289608797837782689905202, 3.02485739121450765925308870027, 3.19909678213323362108365243633, 3.59173199053653741745129210331
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.