Dirichlet series
| L(s) = 1 | + 24·2-s + 216·3-s + 57·4-s + 378·5-s + 5.18e3·6-s + 126·7-s − 2.86e3·8-s + 2.62e4·9-s + 9.07e3·10-s + 6.93e3·11-s + 1.23e4·12-s + 1.24e4·13-s + 3.02e3·14-s + 8.16e4·15-s − 2.83e4·16-s + 2.44e4·17-s + 6.29e5·18-s − 1.46e4·19-s + 2.15e4·20-s + 2.72e4·21-s + 1.66e5·22-s + 9.73e4·23-s − 6.19e5·24-s − 1.68e5·25-s + 2.97e5·26-s + 2.36e6·27-s + 7.18e3·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 4.61·3-s + 0.445·4-s + 1.35·5-s + 9.79·6-s + 0.138·7-s − 1.98·8-s + 12·9-s + 2.86·10-s + 1.57·11-s + 2.05·12-s + 1.56·13-s + 0.294·14-s + 6.24·15-s − 1.72·16-s + 1.20·17-s + 25.4·18-s − 0.491·19-s + 0.602·20-s + 0.641·21-s + 3.33·22-s + 1.66·23-s − 9.14·24-s − 2.15·25-s + 3.32·26-s + 23.0·27-s + 0.0618·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 23^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.65923\times 10^{10}\) |
| Root analytic conductor: | \(4.64268\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 23^{8} ,\ ( \ : [7/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(746.3539541\) |
| \(L(\frac12)\) | \(\approx\) | \(746.3539541\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - p^{3} T )^{8} \) |
| 23 | \( ( 1 - p^{3} T )^{8} \) | |
| good | 2 | \( 1 - 3 p^{3} T + 519 T^{2} - 2055 p^{2} T^{3} + 31795 p^{2} T^{4} - 53861 p^{5} T^{5} + 374505 p^{6} T^{6} - 593891 p^{9} T^{7} + 14289659 p^{8} T^{8} - 593891 p^{16} T^{9} + 374505 p^{20} T^{10} - 53861 p^{26} T^{11} + 31795 p^{30} T^{12} - 2055 p^{37} T^{13} + 519 p^{42} T^{14} - 3 p^{52} T^{15} + p^{56} T^{16} \) |
| 5 | \( 1 - 378 T + 62324 p T^{2} - 119959982 T^{3} + 51012809944 T^{4} - 3526691572738 p T^{5} + 243889110683436 p^{2} T^{6} - 13565056108027894 p^{3} T^{7} + 882987792852762798 p^{4} T^{8} - 13565056108027894 p^{10} T^{9} + 243889110683436 p^{16} T^{10} - 3526691572738 p^{22} T^{11} + 51012809944 p^{28} T^{12} - 119959982 p^{35} T^{13} + 62324 p^{43} T^{14} - 378 p^{49} T^{15} + p^{56} T^{16} \) | |
| 7 | \( 1 - 18 p T + 2329696 T^{2} - 309141710 T^{3} + 2997781433728 T^{4} - 863532935866238 T^{5} + 2705105191557299392 T^{6} - \)\(13\!\cdots\!02\)\( T^{7} + \)\(22\!\cdots\!82\)\( T^{8} - \)\(13\!\cdots\!02\)\( p^{7} T^{9} + 2705105191557299392 p^{14} T^{10} - 863532935866238 p^{21} T^{11} + 2997781433728 p^{28} T^{12} - 309141710 p^{35} T^{13} + 2329696 p^{42} T^{14} - 18 p^{50} T^{15} + p^{56} T^{16} \) | |
| 11 | \( 1 - 6932 T + 101104016 T^{2} - 656906341596 T^{3} + 5394585136104012 T^{4} - 29337433342601668620 T^{5} + \)\(18\!\cdots\!60\)\( T^{6} - \)\(83\!\cdots\!64\)\( T^{7} + \)\(42\!\cdots\!66\)\( T^{8} - \)\(83\!\cdots\!64\)\( p^{7} T^{9} + \)\(18\!\cdots\!60\)\( p^{14} T^{10} - 29337433342601668620 p^{21} T^{11} + 5394585136104012 p^{28} T^{12} - 656906341596 p^{35} T^{13} + 101104016 p^{42} T^{14} - 6932 p^{49} T^{15} + p^{56} T^{16} \) | |
| 13 | \( 1 - 12404 T + 317404680 T^{2} - 3458355434188 T^{3} + 50469360221212396 T^{4} - \)\(46\!\cdots\!56\)\( T^{5} + \)\(52\!\cdots\!92\)\( T^{6} - \)\(41\!\cdots\!20\)\( T^{7} + \)\(38\!\cdots\!10\)\( T^{8} - \)\(41\!\cdots\!20\)\( p^{7} T^{9} + \)\(52\!\cdots\!92\)\( p^{14} T^{10} - \)\(46\!\cdots\!56\)\( p^{21} T^{11} + 50469360221212396 p^{28} T^{12} - 3458355434188 p^{35} T^{13} + 317404680 p^{42} T^{14} - 12404 p^{49} T^{15} + p^{56} T^{16} \) | |
| 17 | \( 1 - 24434 T + 2243476668 T^{2} - 51803192692630 T^{3} + 2414998878753816440 T^{4} - \)\(51\!\cdots\!94\)\( T^{5} + \)\(16\!\cdots\!48\)\( T^{6} - \)\(31\!\cdots\!34\)\( T^{7} + \)\(79\!\cdots\!58\)\( T^{8} - \)\(31\!\cdots\!34\)\( p^{7} T^{9} + \)\(16\!\cdots\!48\)\( p^{14} T^{10} - \)\(51\!\cdots\!94\)\( p^{21} T^{11} + 2414998878753816440 p^{28} T^{12} - 51803192692630 p^{35} T^{13} + 2243476668 p^{42} T^{14} - 24434 p^{49} T^{15} + p^{56} T^{16} \) | |
| 19 | \( 1 + 14682 T + 3079981280 T^{2} + 104879479018762 T^{3} + 5451277498859699424 T^{4} + \)\(21\!\cdots\!34\)\( T^{5} + \)\(83\!\cdots\!64\)\( T^{6} + \)\(25\!\cdots\!78\)\( T^{7} + \)\(92\!\cdots\!02\)\( T^{8} + \)\(25\!\cdots\!78\)\( p^{7} T^{9} + \)\(83\!\cdots\!64\)\( p^{14} T^{10} + \)\(21\!\cdots\!34\)\( p^{21} T^{11} + 5451277498859699424 p^{28} T^{12} + 104879479018762 p^{35} T^{13} + 3079981280 p^{42} T^{14} + 14682 p^{49} T^{15} + p^{56} T^{16} \) | |
| 29 | \( 1 - 255356 T + 52500788912 T^{2} - 10252632912505764 T^{3} + \)\(12\!\cdots\!08\)\( T^{4} - \)\(82\!\cdots\!36\)\( T^{5} + \)\(27\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!80\)\( T^{7} - \)\(29\!\cdots\!30\)\( T^{8} + \)\(12\!\cdots\!80\)\( p^{7} T^{9} + \)\(27\!\cdots\!44\)\( p^{14} T^{10} - \)\(82\!\cdots\!36\)\( p^{21} T^{11} + \)\(12\!\cdots\!08\)\( p^{28} T^{12} - 10252632912505764 p^{35} T^{13} + 52500788912 p^{42} T^{14} - 255356 p^{49} T^{15} + p^{56} T^{16} \) | |
| 31 | \( 1 - 450764 T + 193580523192 T^{2} - 52877331882925740 T^{3} + \)\(14\!\cdots\!56\)\( T^{4} - \)\(30\!\cdots\!76\)\( T^{5} + \)\(66\!\cdots\!40\)\( T^{6} - \)\(11\!\cdots\!68\)\( T^{7} + \)\(21\!\cdots\!38\)\( T^{8} - \)\(11\!\cdots\!68\)\( p^{7} T^{9} + \)\(66\!\cdots\!40\)\( p^{14} T^{10} - \)\(30\!\cdots\!76\)\( p^{21} T^{11} + \)\(14\!\cdots\!56\)\( p^{28} T^{12} - 52877331882925740 p^{35} T^{13} + 193580523192 p^{42} T^{14} - 450764 p^{49} T^{15} + p^{56} T^{16} \) | |
| 37 | \( 1 - 206240 T + 319562149936 T^{2} - 91400733922971672 T^{3} + \)\(56\!\cdots\!72\)\( T^{4} - \)\(19\!\cdots\!44\)\( T^{5} + \)\(77\!\cdots\!80\)\( T^{6} - \)\(25\!\cdots\!00\)\( T^{7} + \)\(84\!\cdots\!94\)\( T^{8} - \)\(25\!\cdots\!00\)\( p^{7} T^{9} + \)\(77\!\cdots\!80\)\( p^{14} T^{10} - \)\(19\!\cdots\!44\)\( p^{21} T^{11} + \)\(56\!\cdots\!72\)\( p^{28} T^{12} - 91400733922971672 p^{35} T^{13} + 319562149936 p^{42} T^{14} - 206240 p^{49} T^{15} + p^{56} T^{16} \) | |
| 41 | \( 1 - 1053344 T + 1370061910600 T^{2} - 868206254256653600 T^{3} + \)\(68\!\cdots\!60\)\( T^{4} - \)\(33\!\cdots\!80\)\( T^{5} + \)\(21\!\cdots\!16\)\( T^{6} - \)\(94\!\cdots\!08\)\( T^{7} + \)\(51\!\cdots\!70\)\( T^{8} - \)\(94\!\cdots\!08\)\( p^{7} T^{9} + \)\(21\!\cdots\!16\)\( p^{14} T^{10} - \)\(33\!\cdots\!80\)\( p^{21} T^{11} + \)\(68\!\cdots\!60\)\( p^{28} T^{12} - 868206254256653600 p^{35} T^{13} + 1370061910600 p^{42} T^{14} - 1053344 p^{49} T^{15} + p^{56} T^{16} \) | |
| 43 | \( 1 - 1587806 T + 2475445659440 T^{2} - 2550109454356663182 T^{3} + \)\(24\!\cdots\!80\)\( T^{4} - \)\(18\!\cdots\!10\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(78\!\cdots\!58\)\( T^{7} + \)\(43\!\cdots\!42\)\( T^{8} - \)\(78\!\cdots\!58\)\( p^{7} T^{9} + \)\(12\!\cdots\!56\)\( p^{14} T^{10} - \)\(18\!\cdots\!10\)\( p^{21} T^{11} + \)\(24\!\cdots\!80\)\( p^{28} T^{12} - 2550109454356663182 p^{35} T^{13} + 2475445659440 p^{42} T^{14} - 1587806 p^{49} T^{15} + p^{56} T^{16} \) | |
| 47 | \( 1 - 443336 T + 1714617247408 T^{2} - 22554013848427112 T^{3} + \)\(13\!\cdots\!00\)\( T^{4} + \)\(47\!\cdots\!56\)\( T^{5} + \)\(69\!\cdots\!24\)\( T^{6} + \)\(53\!\cdots\!68\)\( T^{7} + \)\(31\!\cdots\!54\)\( T^{8} + \)\(53\!\cdots\!68\)\( p^{7} T^{9} + \)\(69\!\cdots\!24\)\( p^{14} T^{10} + \)\(47\!\cdots\!56\)\( p^{21} T^{11} + \)\(13\!\cdots\!00\)\( p^{28} T^{12} - 22554013848427112 p^{35} T^{13} + 1714617247408 p^{42} T^{14} - 443336 p^{49} T^{15} + p^{56} T^{16} \) | |
| 53 | \( 1 + 375530 T + 4190547027244 T^{2} + 3281441479222057966 T^{3} + \)\(18\!\cdots\!52\)\( p T^{4} + \)\(98\!\cdots\!58\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!38\)\( T^{7} + \)\(21\!\cdots\!02\)\( T^{8} + \)\(18\!\cdots\!38\)\( p^{7} T^{9} + \)\(16\!\cdots\!72\)\( p^{14} T^{10} + \)\(98\!\cdots\!58\)\( p^{21} T^{11} + \)\(18\!\cdots\!52\)\( p^{29} T^{12} + 3281441479222057966 p^{35} T^{13} + 4190547027244 p^{42} T^{14} + 375530 p^{49} T^{15} + p^{56} T^{16} \) | |
| 59 | \( 1 - 624008 T + 12956170143360 T^{2} - 9529541625120889672 T^{3} + \)\(84\!\cdots\!60\)\( T^{4} - \)\(62\!\cdots\!04\)\( T^{5} + \)\(35\!\cdots\!40\)\( T^{6} - \)\(23\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!98\)\( T^{8} - \)\(23\!\cdots\!40\)\( p^{7} T^{9} + \)\(35\!\cdots\!40\)\( p^{14} T^{10} - \)\(62\!\cdots\!04\)\( p^{21} T^{11} + \)\(84\!\cdots\!60\)\( p^{28} T^{12} - 9529541625120889672 p^{35} T^{13} + 12956170143360 p^{42} T^{14} - 624008 p^{49} T^{15} + p^{56} T^{16} \) | |
| 61 | \( 1 + 2005568 T + 12095059800288 T^{2} + 31113913488939614360 T^{3} + \)\(87\!\cdots\!04\)\( T^{4} + \)\(20\!\cdots\!24\)\( T^{5} + \)\(46\!\cdots\!72\)\( T^{6} + \)\(87\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!26\)\( T^{8} + \)\(87\!\cdots\!20\)\( p^{7} T^{9} + \)\(46\!\cdots\!72\)\( p^{14} T^{10} + \)\(20\!\cdots\!24\)\( p^{21} T^{11} + \)\(87\!\cdots\!04\)\( p^{28} T^{12} + 31113913488939614360 p^{35} T^{13} + 12095059800288 p^{42} T^{14} + 2005568 p^{49} T^{15} + p^{56} T^{16} \) | |
| 67 | \( 1 + 2712286 T + 27199761811120 T^{2} + 59135267254770991038 T^{3} + \)\(34\!\cdots\!88\)\( T^{4} + \)\(64\!\cdots\!06\)\( T^{5} + \)\(29\!\cdots\!48\)\( T^{6} + \)\(50\!\cdots\!18\)\( T^{7} + \)\(20\!\cdots\!42\)\( T^{8} + \)\(50\!\cdots\!18\)\( p^{7} T^{9} + \)\(29\!\cdots\!48\)\( p^{14} T^{10} + \)\(64\!\cdots\!06\)\( p^{21} T^{11} + \)\(34\!\cdots\!88\)\( p^{28} T^{12} + 59135267254770991038 p^{35} T^{13} + 27199761811120 p^{42} T^{14} + 2712286 p^{49} T^{15} + p^{56} T^{16} \) | |
| 71 | \( 1 + 6287176 T + 48800931783528 T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!64\)\( T^{4} + \)\(40\!\cdots\!72\)\( T^{5} + \)\(16\!\cdots\!80\)\( T^{6} + \)\(52\!\cdots\!48\)\( T^{7} + \)\(18\!\cdots\!66\)\( T^{8} + \)\(52\!\cdots\!48\)\( p^{7} T^{9} + \)\(16\!\cdots\!80\)\( p^{14} T^{10} + \)\(40\!\cdots\!72\)\( p^{21} T^{11} + \)\(11\!\cdots\!64\)\( p^{28} T^{12} + \)\(21\!\cdots\!48\)\( p^{35} T^{13} + 48800931783528 p^{42} T^{14} + 6287176 p^{49} T^{15} + p^{56} T^{16} \) | |
| 73 | \( 1 + 10358312 T + 106805898338280 T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!24\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(92\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} + \)\(34\!\cdots\!12\)\( p^{7} T^{9} + \)\(92\!\cdots\!36\)\( p^{14} T^{10} + \)\(20\!\cdots\!60\)\( p^{21} T^{11} + \)\(42\!\cdots\!24\)\( p^{28} T^{12} + \)\(68\!\cdots\!40\)\( p^{35} T^{13} + 106805898338280 p^{42} T^{14} + 10358312 p^{49} T^{15} + p^{56} T^{16} \) | |
| 79 | \( 1 + 8800574 T + 129662512342512 T^{2} + \)\(81\!\cdots\!94\)\( T^{3} + \)\(68\!\cdots\!04\)\( T^{4} + \)\(33\!\cdots\!86\)\( T^{5} + \)\(21\!\cdots\!96\)\( T^{6} + \)\(86\!\cdots\!34\)\( T^{7} + \)\(46\!\cdots\!54\)\( T^{8} + \)\(86\!\cdots\!34\)\( p^{7} T^{9} + \)\(21\!\cdots\!96\)\( p^{14} T^{10} + \)\(33\!\cdots\!86\)\( p^{21} T^{11} + \)\(68\!\cdots\!04\)\( p^{28} T^{12} + \)\(81\!\cdots\!94\)\( p^{35} T^{13} + 129662512342512 p^{42} T^{14} + 8800574 p^{49} T^{15} + p^{56} T^{16} \) | |
| 83 | \( 1 - 384948 T + 169821921035856 T^{2} - 69870542061003941036 T^{3} + \)\(13\!\cdots\!40\)\( T^{4} - \)\(54\!\cdots\!60\)\( T^{5} + \)\(66\!\cdots\!72\)\( T^{6} - \)\(24\!\cdots\!88\)\( T^{7} + \)\(21\!\cdots\!22\)\( T^{8} - \)\(24\!\cdots\!88\)\( p^{7} T^{9} + \)\(66\!\cdots\!72\)\( p^{14} T^{10} - \)\(54\!\cdots\!60\)\( p^{21} T^{11} + \)\(13\!\cdots\!40\)\( p^{28} T^{12} - 69870542061003941036 p^{35} T^{13} + 169821921035856 p^{42} T^{14} - 384948 p^{49} T^{15} + p^{56} T^{16} \) | |
| 89 | \( 1 + 3445530 T + 82600338908068 T^{2} + \)\(67\!\cdots\!34\)\( T^{3} + \)\(45\!\cdots\!32\)\( T^{4} + \)\(38\!\cdots\!70\)\( T^{5} + \)\(26\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!42\)\( T^{7} + \)\(12\!\cdots\!66\)\( T^{8} + \)\(14\!\cdots\!42\)\( p^{7} T^{9} + \)\(26\!\cdots\!16\)\( p^{14} T^{10} + \)\(38\!\cdots\!70\)\( p^{21} T^{11} + \)\(45\!\cdots\!32\)\( p^{28} T^{12} + \)\(67\!\cdots\!34\)\( p^{35} T^{13} + 82600338908068 p^{42} T^{14} + 3445530 p^{49} T^{15} + p^{56} T^{16} \) | |
| 97 | \( 1 + 28043764 T + 829820168627104 T^{2} + \)\(15\!\cdots\!76\)\( T^{3} + \)\(26\!\cdots\!48\)\( T^{4} + \)\(34\!\cdots\!96\)\( T^{5} + \)\(44\!\cdots\!44\)\( T^{6} + \)\(46\!\cdots\!04\)\( T^{7} + \)\(45\!\cdots\!22\)\( T^{8} + \)\(46\!\cdots\!04\)\( p^{7} T^{9} + \)\(44\!\cdots\!44\)\( p^{14} T^{10} + \)\(34\!\cdots\!96\)\( p^{21} T^{11} + \)\(26\!\cdots\!48\)\( p^{28} T^{12} + \)\(15\!\cdots\!76\)\( p^{35} T^{13} + 829820168627104 p^{42} T^{14} + 28043764 p^{49} T^{15} + p^{56} T^{16} \) | |
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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
−5.35871731897374510465462271755, −4.76438724005991075317311137895, −4.72867406275401722125981384351, −4.50485095219149702262254865755, −4.41803420991700869600457114297, −4.29109766510332481824571749337, −4.19928359499263201926534822281, −4.17275803961446509530753376951, −4.12392635385709258768362526194, −3.47266567310524062709461960052, −3.29516518957632270922038236763, −3.16155955781121281193469615631, −3.13606973911421250556285113670, −3.06209934128242571816145236537, −2.66847044944035791687862439662, −2.62916581769461796212104820489, −2.22168486101543150311364625944, −1.93978661155807012447190946587, −1.78269142561771628383591720792, −1.71519289393176200647452926486, −1.23185850544267533210518505911, −1.09972947380201448561963309220, −1.04312081132085226704607552122, −0.75756086151303984421225031355, −0.37616687599646011118164969407, 0.37616687599646011118164969407, 0.75756086151303984421225031355, 1.04312081132085226704607552122, 1.09972947380201448561963309220, 1.23185850544267533210518505911, 1.71519289393176200647452926486, 1.78269142561771628383591720792, 1.93978661155807012447190946587, 2.22168486101543150311364625944, 2.62916581769461796212104820489, 2.66847044944035791687862439662, 3.06209934128242571816145236537, 3.13606973911421250556285113670, 3.16155955781121281193469615631, 3.29516518957632270922038236763, 3.47266567310524062709461960052, 4.12392635385709258768362526194, 4.17275803961446509530753376951, 4.19928359499263201926534822281, 4.29109766510332481824571749337, 4.41803420991700869600457114297, 4.50485095219149702262254865755, 4.72867406275401722125981384351, 4.76438724005991075317311137895, 5.35871731897374510465462271755
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.