Dirichlet series
| L(s) = 1 | − 256·2-s − 2.91e3·3-s + 4.09e4·4-s + 1.73e4·5-s + 7.46e5·6-s + 1.10e5·7-s − 5.24e6·8-s + 5.31e6·9-s − 4.42e6·10-s + 5.02e6·11-s − 1.19e8·12-s + 1.93e7·13-s − 2.83e7·14-s − 5.04e7·15-s + 5.87e8·16-s + 1.78e8·17-s − 1.36e9·18-s + 3.30e8·19-s + 7.08e8·20-s − 3.23e8·21-s − 1.28e9·22-s + 4.83e8·23-s + 1.52e10·24-s − 1.31e9·25-s − 4.94e9·26-s − 7.74e9·27-s + 4.53e9·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 0.495·5-s + 6.53·6-s + 0.355·7-s − 7.07·8-s + 10/3·9-s − 1.40·10-s + 0.855·11-s − 11.5·12-s + 1.10·13-s − 1.00·14-s − 1.14·15-s + 35/4·16-s + 1.79·17-s − 9.42·18-s + 1.61·19-s + 2.47·20-s − 0.821·21-s − 2.42·22-s + 0.681·23-s + 16.3·24-s − 1.08·25-s − 3.13·26-s − 3.84·27-s + 1.77·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.89394\times 10^{7}\) |
| Root analytic conductor: | \(9.14549\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 37015056,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(1.002699974\) |
| \(L(\frac12)\) | \(\approx\) | \(1.002699974\) |
| \(L(\frac{15}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 17302 T + 1618921176 T^{2} - 2450744796642 p^{2} T^{3} + 4457772480590782 p^{4} T^{4} - 2450744796642 p^{15} T^{5} + 1618921176 p^{26} T^{6} - 17302 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 110778 T + 276829359572 T^{2} - 4931357542986702 p T^{3} + \)\(71\!\cdots\!06\)\( p^{2} T^{4} - 4931357542986702 p^{14} T^{5} + 276829359572 p^{26} T^{6} - 110778 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 5029228 T + 1875262762644 p T^{2} - 440532243269992908 p^{2} T^{3} + \)\(37\!\cdots\!22\)\( p^{3} T^{4} - 440532243269992908 p^{15} T^{5} + 1875262762644 p^{27} T^{6} - 5029228 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 178554136 T + 23824026730331580 T^{2} - \)\(21\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!32\)\( p^{13} T^{5} + 23824026730331580 p^{26} T^{6} - 178554136 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 330495078 T + 189176579634440540 T^{2} - \)\(41\!\cdots\!22\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(41\!\cdots\!22\)\( p^{13} T^{5} + 189176579634440540 p^{26} T^{6} - 330495078 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 483904200 T + 40056200449243780 p T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!62\)\( T^{4} - \)\(43\!\cdots\!40\)\( p^{13} T^{5} + 40056200449243780 p^{27} T^{6} - 483904200 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 3450624948 T + 28398170401550067428 T^{2} - \)\(82\!\cdots\!28\)\( T^{3} + \)\(42\!\cdots\!34\)\( T^{4} - \)\(82\!\cdots\!28\)\( p^{13} T^{5} + 28398170401550067428 p^{26} T^{6} - 3450624948 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 1760560078 T - 14782363854974689100 T^{2} - \)\(11\!\cdots\!54\)\( p T^{3} + \)\(10\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!54\)\( p^{14} T^{5} - 14782363854974689100 p^{26} T^{6} + 1760560078 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 21294929844 T + \)\(88\!\cdots\!48\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(31\!\cdots\!42\)\( T^{4} + \)\(13\!\cdots\!56\)\( p^{13} T^{5} + \)\(88\!\cdots\!48\)\( p^{26} T^{6} + 21294929844 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 70606057250 T + \)\(47\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!02\)\( T^{3} + \)\(68\!\cdots\!78\)\( T^{4} + \)\(19\!\cdots\!02\)\( p^{13} T^{5} + \)\(47\!\cdots\!00\)\( p^{26} T^{6} + 70606057250 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 75558364632 T + \)\(58\!\cdots\!80\)\( T^{2} + \)\(20\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(20\!\cdots\!96\)\( p^{13} T^{5} + \)\(58\!\cdots\!80\)\( p^{26} T^{6} + 75558364632 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 112183555592 T + \)\(14\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!86\)\( T^{4} + \)\(13\!\cdots\!20\)\( p^{13} T^{5} + \)\(14\!\cdots\!32\)\( p^{26} T^{6} + 112183555592 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 104575883144 T - \)\(20\!\cdots\!84\)\( T^{2} + \)\(54\!\cdots\!64\)\( T^{3} + \)\(74\!\cdots\!18\)\( T^{4} + \)\(54\!\cdots\!64\)\( p^{13} T^{5} - \)\(20\!\cdots\!84\)\( p^{26} T^{6} + 104575883144 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 953392600312 T + \)\(50\!\cdots\!04\)\( T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(65\!\cdots\!90\)\( T^{4} + \)\(19\!\cdots\!76\)\( p^{13} T^{5} + \)\(50\!\cdots\!04\)\( p^{26} T^{6} + 953392600312 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 173990512884 T + \)\(55\!\cdots\!36\)\( T^{2} + \)\(58\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} + \)\(58\!\cdots\!32\)\( p^{13} T^{5} + \)\(55\!\cdots\!36\)\( p^{26} T^{6} + 173990512884 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 163049862070 T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!30\)\( T^{3} + \)\(65\!\cdots\!34\)\( T^{4} + \)\(33\!\cdots\!30\)\( p^{13} T^{5} + \)\(10\!\cdots\!00\)\( p^{26} T^{6} + 163049862070 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 1141535539628 T + \)\(90\!\cdots\!44\)\( T^{2} - \)\(96\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!42\)\( T^{4} - \)\(96\!\cdots\!88\)\( p^{13} T^{5} + \)\(90\!\cdots\!44\)\( p^{26} T^{6} - 1141535539628 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 2505023734484 T + \)\(61\!\cdots\!08\)\( T^{2} - \)\(88\!\cdots\!52\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} - \)\(88\!\cdots\!52\)\( p^{13} T^{5} + \)\(61\!\cdots\!08\)\( p^{26} T^{6} - 2505023734484 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 2114849719736 T + \)\(13\!\cdots\!04\)\( T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(88\!\cdots\!42\)\( T^{4} - \)\(28\!\cdots\!48\)\( p^{13} T^{5} + \)\(13\!\cdots\!04\)\( p^{26} T^{6} - 2114849719736 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 3245912192784 T + \)\(18\!\cdots\!00\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(23\!\cdots\!20\)\( p^{13} T^{5} + \)\(18\!\cdots\!00\)\( p^{26} T^{6} - 3245912192784 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 4151539197470 T + \)\(74\!\cdots\!48\)\( T^{2} + \)\(20\!\cdots\!10\)\( T^{3} + \)\(22\!\cdots\!74\)\( T^{4} + \)\(20\!\cdots\!10\)\( p^{13} T^{5} + \)\(74\!\cdots\!48\)\( p^{26} T^{6} + 4151539197470 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 7366767632100 T + \)\(23\!\cdots\!12\)\( T^{2} - \)\(15\!\cdots\!96\)\( T^{3} + \)\(22\!\cdots\!50\)\( T^{4} - \)\(15\!\cdots\!96\)\( p^{13} T^{5} + \)\(23\!\cdots\!12\)\( p^{26} T^{6} - 7366767632100 p^{39} T^{7} + p^{52} T^{8} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.229909415117453865471207308484, −7.59132239421658701567945071098, −7.46000340836011889670178749068, −7.25055869045576836556705129064, −6.96132975206025962684221359593, −6.28060780356101340721383013727, −6.23420287235228959110344343440, −6.21496381636578006768515253411, −6.11523496463524049025456729120, −5.20921716372742629263067491114, −5.17107762425573644909486811357, −4.80772237272434038436374066212, −4.79935500677740560387012310023, −3.55685917349594796647715849134, −3.48152141592082440843133460920, −3.22243815499382929130189440090, −3.18091871113193761214924151776, −1.93955254156629943868779643693, −1.80952798861572595438691266330, −1.59496018176762590640264426911, −1.58207069429771095163728461397, −1.03606032427052991663531058258, −0.74756776002976782776570343305, −0.46835355708027141779802158812, −0.36197664366443949794885128016, 0.36197664366443949794885128016, 0.46835355708027141779802158812, 0.74756776002976782776570343305, 1.03606032427052991663531058258, 1.58207069429771095163728461397, 1.59496018176762590640264426911, 1.80952798861572595438691266330, 1.93955254156629943868779643693, 3.18091871113193761214924151776, 3.22243815499382929130189440090, 3.48152141592082440843133460920, 3.55685917349594796647715849134, 4.79935500677740560387012310023, 4.80772237272434038436374066212, 5.17107762425573644909486811357, 5.20921716372742629263067491114, 6.11523496463524049025456729120, 6.21496381636578006768515253411, 6.23420287235228959110344343440, 6.28060780356101340721383013727, 6.96132975206025962684221359593, 7.25055869045576836556705129064, 7.46000340836011889670178749068, 7.59132239421658701567945071098, 8.229909415117453865471207308484