Dirichlet series
| L(s) = 1 | − 512·4-s + 1.49e3·7-s + 4.67e4·13-s + 1.63e5·16-s − 6.40e5·19-s + 1.48e6·25-s − 7.63e5·28-s + 1.07e5·31-s − 1.45e6·37-s − 8.46e6·43-s − 9.99e6·49-s − 2.39e7·52-s − 2.83e7·61-s − 4.19e7·64-s + 9.30e7·67-s − 1.39e8·73-s + 3.27e8·76-s − 4.08e7·79-s + 6.97e7·91-s − 4.94e7·97-s − 7.59e8·100-s + 6.41e8·103-s − 2.01e8·109-s + 2.44e8·112-s + 1.62e9·121-s − 5.51e7·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.621·7-s + 1.63·13-s + 5/2·16-s − 4.91·19-s + 3.79·25-s − 1.24·28-s + 0.116·31-s − 0.778·37-s − 2.47·43-s − 1.73·49-s − 3.27·52-s − 2.04·61-s − 5/2·64-s + 4.61·67-s − 4.90·73-s + 9.82·76-s − 1.04·79-s + 1.01·91-s − 0.558·97-s − 7.59·100-s + 5.70·103-s − 1.42·109-s + 1.55·112-s + 7.59·121-s − 0.233·124-s − 3.05·133-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.59837\times 10^{14}\) |
| Root analytic conductor: | \(8.12375\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{32} ,\ ( \ : [4]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.7156844018\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7156844018\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - 1484192 T^{2} + 943312779166 T^{4} - 14130474162592256 p^{2} T^{6} + \)\(18\!\cdots\!91\)\( p^{4} T^{8} - 14130474162592256 p^{18} T^{10} + 943312779166 p^{32} T^{12} - 1484192 p^{48} T^{14} + p^{64} T^{16} \) |
| 7 | \( ( 1 - 746 T + 5830042 T^{2} + 2377297702 p T^{3} + 162229945954 p^{2} T^{4} + 2377297702 p^{9} T^{5} + 5830042 p^{16} T^{6} - 746 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 11 | \( 1 - 1627905020 T^{2} + 1177304987282645464 T^{4} - \)\(49\!\cdots\!44\)\( T^{6} + \)\(10\!\cdots\!94\)\( p^{2} T^{8} - \)\(49\!\cdots\!44\)\( p^{16} T^{10} + 1177304987282645464 p^{32} T^{12} - 1627905020 p^{48} T^{14} + p^{64} T^{16} \) | |
| 13 | \( ( 1 - 23390 T + 1522306078 T^{2} - 37174745131328 T^{3} + 1826529682359656563 T^{4} - 37174745131328 p^{8} T^{5} + 1522306078 p^{16} T^{6} - 23390 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 17 | \( 1 - 5344391960 T^{2} + 67014418685538964294 T^{4} - \)\(22\!\cdots\!36\)\( p T^{6} + \)\(20\!\cdots\!79\)\( T^{8} - \)\(22\!\cdots\!36\)\( p^{17} T^{10} + 67014418685538964294 p^{32} T^{12} - 5344391960 p^{48} T^{14} + p^{64} T^{16} \) | |
| 19 | \( ( 1 + 320218 T + 93914072434 T^{2} + 16671870506023270 T^{3} + \)\(26\!\cdots\!22\)\( T^{4} + 16671870506023270 p^{8} T^{5} + 93914072434 p^{16} T^{6} + 320218 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 23 | \( 1 - 403847606900 T^{2} + \)\(78\!\cdots\!56\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{6} + \)\(87\!\cdots\!26\)\( T^{8} - \)\(96\!\cdots\!00\)\( p^{16} T^{10} + \)\(78\!\cdots\!56\)\( p^{32} T^{12} - 403847606900 p^{48} T^{14} + p^{64} T^{16} \) | |
| 29 | \( 1 - 2672023629992 T^{2} + \)\(35\!\cdots\!22\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!39\)\( T^{8} - \)\(30\!\cdots\!40\)\( p^{16} T^{10} + \)\(35\!\cdots\!22\)\( p^{32} T^{12} - 2672023629992 p^{48} T^{14} + p^{64} T^{16} \) | |
| 31 | \( ( 1 - 53888 T + 2368364178904 T^{2} - 153158519194135616 T^{3} + \)\(27\!\cdots\!02\)\( T^{4} - 153158519194135616 p^{8} T^{5} + 2368364178904 p^{16} T^{6} - 53888 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 729562 T + 9450239831902 T^{2} + 6123825829818334888 T^{3} + \)\(43\!\cdots\!55\)\( T^{4} + 6123825829818334888 p^{8} T^{5} + 9450239831902 p^{16} T^{6} + 729562 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 41 | \( 1 - 639826802392 p T^{2} + \)\(45\!\cdots\!80\)\( T^{4} - \)\(56\!\cdots\!00\)\( T^{6} + \)\(51\!\cdots\!26\)\( T^{8} - \)\(56\!\cdots\!00\)\( p^{16} T^{10} + \)\(45\!\cdots\!80\)\( p^{32} T^{12} - 639826802392 p^{49} T^{14} + p^{64} T^{16} \) | |
| 43 | \( ( 1 + 4232734 T + 44347538440786 T^{2} + \)\(12\!\cdots\!58\)\( T^{3} + \)\(73\!\cdots\!78\)\( T^{4} + \)\(12\!\cdots\!58\)\( p^{8} T^{5} + 44347538440786 p^{16} T^{6} + 4232734 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 47 | \( 1 - 52269673246232 T^{2} + \)\(22\!\cdots\!08\)\( T^{4} - \)\(67\!\cdots\!16\)\( T^{6} + \)\(17\!\cdots\!54\)\( T^{8} - \)\(67\!\cdots\!16\)\( p^{16} T^{10} + \)\(22\!\cdots\!08\)\( p^{32} T^{12} - 52269673246232 p^{48} T^{14} + p^{64} T^{16} \) | |
| 53 | \( 1 - 479373452354168 T^{2} + \)\(10\!\cdots\!52\)\( T^{4} - \)\(12\!\cdots\!72\)\( T^{6} + \)\(96\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!72\)\( p^{16} T^{10} + \)\(10\!\cdots\!52\)\( p^{32} T^{12} - 479373452354168 p^{48} T^{14} + p^{64} T^{16} \) | |
| 59 | \( 1 - 375445675770104 T^{2} + \)\(10\!\cdots\!44\)\( T^{4} - \)\(21\!\cdots\!92\)\( T^{6} + \)\(37\!\cdots\!78\)\( T^{8} - \)\(21\!\cdots\!92\)\( p^{16} T^{10} + \)\(10\!\cdots\!44\)\( p^{32} T^{12} - 375445675770104 p^{48} T^{14} + p^{64} T^{16} \) | |
| 61 | \( ( 1 + 14168698 T + 164344606366738 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} - \)\(22\!\cdots\!49\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{8} T^{5} + 164344606366738 p^{16} T^{6} + 14168698 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 46506398 T + 1882083225120874 T^{2} - \)\(49\!\cdots\!10\)\( T^{3} + \)\(12\!\cdots\!62\)\( T^{4} - \)\(49\!\cdots\!10\)\( p^{8} T^{5} + 1882083225120874 p^{16} T^{6} - 46506398 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 71 | \( 1 - 2494655249307668 T^{2} + \)\(32\!\cdots\!04\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{6} + \)\(21\!\cdots\!98\)\( T^{8} - \)\(29\!\cdots\!24\)\( p^{16} T^{10} + \)\(32\!\cdots\!04\)\( p^{32} T^{12} - 2494655249307668 p^{48} T^{14} + p^{64} T^{16} \) | |
| 73 | \( ( 1 + 69655456 T + 4269139451624278 T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!03\)\( T^{4} + \)\(16\!\cdots\!00\)\( p^{8} T^{5} + 4269139451624278 p^{16} T^{6} + 69655456 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 79 | \( ( 1 + 20445178 T + 3204890336672626 T^{2} + \)\(27\!\cdots\!10\)\( T^{3} + \)\(57\!\cdots\!14\)\( T^{4} + \)\(27\!\cdots\!10\)\( p^{8} T^{5} + 3204890336672626 p^{16} T^{6} + 20445178 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
| 83 | \( 1 - 10156350284453048 T^{2} + \)\(54\!\cdots\!24\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{6} + \)\(52\!\cdots\!62\)\( T^{8} - \)\(19\!\cdots\!80\)\( p^{16} T^{10} + \)\(54\!\cdots\!24\)\( p^{32} T^{12} - 10156350284453048 p^{48} T^{14} + p^{64} T^{16} \) | |
| 89 | \( 1 - 1642084998993584 T^{2} + \)\(48\!\cdots\!50\)\( T^{4} - \)\(54\!\cdots\!44\)\( T^{6} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(54\!\cdots\!44\)\( p^{16} T^{10} + \)\(48\!\cdots\!50\)\( p^{32} T^{12} - 1642084998993584 p^{48} T^{14} + p^{64} T^{16} \) | |
| 97 | \( ( 1 + 24721456 T + 13466542735344856 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(68\!\cdots\!98\)\( T^{4} + \)\(16\!\cdots\!32\)\( p^{8} T^{5} + 13466542735344856 p^{16} T^{6} + 24721456 p^{24} T^{7} + p^{32} T^{8} )^{2} \) | |
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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
−4.37938477216172157123336647683, −4.34366795655564934734535137688, −4.08970941447721909290640525512, −3.73919476525340847991938883342, −3.59166452564511306725314066230, −3.55015821216792974509968383833, −3.36094359127360709465467373376, −3.25602372303283301887783910926, −3.05907065634471169685245607170, −2.93167613664243315695794871481, −2.65054095265897277227928668105, −2.48825410338387711311823756410, −2.13780104877786507172367083126, −1.98627614048745761011455262615, −1.88071541165050132515992762921, −1.82663042208954804105102056105, −1.56545331897758867596290250834, −1.30726221014609398320314698234, −1.26639408294501610635104698237, −0.979417769342314960642429878261, −0.70562244919557177925797806444, −0.62468835304352853776145790036, −0.46995735853017388047014110242, −0.28630093213847564067190081418, −0.07304949364285546585615977343, 0.07304949364285546585615977343, 0.28630093213847564067190081418, 0.46995735853017388047014110242, 0.62468835304352853776145790036, 0.70562244919557177925797806444, 0.979417769342314960642429878261, 1.26639408294501610635104698237, 1.30726221014609398320314698234, 1.56545331897758867596290250834, 1.82663042208954804105102056105, 1.88071541165050132515992762921, 1.98627614048745761011455262615, 2.13780104877786507172367083126, 2.48825410338387711311823756410, 2.65054095265897277227928668105, 2.93167613664243315695794871481, 3.05907065634471169685245607170, 3.25602372303283301887783910926, 3.36094359127360709465467373376, 3.55015821216792974509968383833, 3.59166452564511306725314066230, 3.73919476525340847991938883342, 4.08970941447721909290640525512, 4.34366795655564934734535137688, 4.37938477216172157123336647683
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.