Dirichlet series
| L(s) = 1 | + 1.02e3·2-s − 2.62e4·3-s + 6.55e5·4-s − 5.18e5·5-s − 2.68e7·6-s − 1.68e7·7-s + 3.35e8·8-s + 4.30e8·9-s − 5.31e8·10-s − 5.49e8·11-s − 1.71e10·12-s + 3.26e9·13-s − 1.72e10·14-s + 1.36e10·15-s + 1.50e11·16-s − 1.69e10·17-s + 4.40e11·18-s − 1.07e10·19-s − 3.39e11·20-s + 4.41e11·21-s − 5.63e11·22-s − 2.61e11·23-s − 8.80e12·24-s − 1.21e12·25-s + 3.34e12·26-s − 5.64e12·27-s − 1.10e13·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 0.593·5-s − 6.53·6-s − 1.10·7-s + 7.07·8-s + 10/3·9-s − 1.67·10-s − 0.773·11-s − 11.5·12-s + 1.10·13-s − 3.12·14-s + 1.37·15-s + 35/4·16-s − 0.590·17-s + 9.42·18-s − 0.145·19-s − 2.96·20-s + 2.54·21-s − 2.18·22-s − 0.695·23-s − 16.3·24-s − 1.59·25-s + 3.13·26-s − 3.84·27-s − 5.51·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(37015056\) = \(2^{4} \cdot 3^{4} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.17147\times 10^{8}\) |
| Root analytic conductor: | \(11.9546\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 37015056,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
Particular Values
| \(L(9)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{19}{2})\) | 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 | $C_1$ | \( ( 1 - p^{8} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - p^{8} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 518584 T + 297193428288 p T^{2} + 654969118627368 p^{4} T^{3} + \)\(42\!\cdots\!22\)\( p^{5} T^{4} + 654969118627368 p^{21} T^{5} + 297193428288 p^{35} T^{6} + 518584 p^{51} T^{7} + p^{68} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 16826340 T + 109565570892656 p T^{2} + \)\(19\!\cdots\!80\)\( p^{2} T^{3} + \)\(10\!\cdots\!90\)\( p^{4} T^{4} + \)\(19\!\cdots\!80\)\( p^{19} T^{5} + 109565570892656 p^{35} T^{6} + 16826340 p^{51} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 549884020 T + 1352826910584371676 T^{2} + \)\(34\!\cdots\!08\)\( p T^{3} + \)\(66\!\cdots\!90\)\( p^{2} T^{4} + \)\(34\!\cdots\!08\)\( p^{18} T^{5} + 1352826910584371676 p^{34} T^{6} + 549884020 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 16988107960 T + \)\(13\!\cdots\!80\)\( T^{2} + \)\(22\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!22\)\( T^{4} + \)\(22\!\cdots\!08\)\( p^{17} T^{5} + \)\(13\!\cdots\!80\)\( p^{34} T^{6} + 16988107960 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 10774690044 T + \)\(92\!\cdots\!08\)\( T^{2} + \)\(38\!\cdots\!24\)\( T^{3} + \)\(61\!\cdots\!74\)\( T^{4} + \)\(38\!\cdots\!24\)\( p^{17} T^{5} + \)\(92\!\cdots\!08\)\( p^{34} T^{6} + 10774690044 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 261020574360 T + \)\(32\!\cdots\!52\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!58\)\( T^{4} + \)\(11\!\cdots\!00\)\( p^{17} T^{5} + \)\(32\!\cdots\!52\)\( p^{34} T^{6} + 261020574360 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 5916347212968 T + \)\(31\!\cdots\!24\)\( T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!04\)\( p^{17} T^{5} + \)\(31\!\cdots\!24\)\( p^{34} T^{6} - 5916347212968 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 653877384884 T + \)\(38\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} + \)\(13\!\cdots\!52\)\( p^{17} T^{5} + \)\(38\!\cdots\!32\)\( p^{34} T^{6} - 653877384884 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 22648713688872 T + \)\(48\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!82\)\( T^{4} - \)\(14\!\cdots\!92\)\( p^{17} T^{5} + \)\(48\!\cdots\!40\)\( p^{34} T^{6} - 22648713688872 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 16695856104776 T + \)\(30\!\cdots\!40\)\( T^{2} + \)\(41\!\cdots\!96\)\( T^{3} + \)\(57\!\cdots\!54\)\( T^{4} + \)\(41\!\cdots\!96\)\( p^{17} T^{5} + \)\(30\!\cdots\!40\)\( p^{34} T^{6} - 16695856104776 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 131852373982632 T + \)\(23\!\cdots\!20\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(20\!\cdots\!50\)\( T^{4} + \)\(19\!\cdots\!68\)\( p^{17} T^{5} + \)\(23\!\cdots\!20\)\( p^{34} T^{6} + 131852373982632 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 392280969756004 T + \)\(15\!\cdots\!32\)\( T^{2} + \)\(34\!\cdots\!04\)\( T^{3} + \)\(71\!\cdots\!30\)\( T^{4} + \)\(34\!\cdots\!04\)\( p^{17} T^{5} + \)\(15\!\cdots\!32\)\( p^{34} T^{6} + 392280969756004 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 74097065727976 T + \)\(21\!\cdots\!92\)\( T^{2} + \)\(25\!\cdots\!64\)\( T^{3} + \)\(10\!\cdots\!54\)\( T^{4} + \)\(25\!\cdots\!64\)\( p^{17} T^{5} + \)\(21\!\cdots\!92\)\( p^{34} T^{6} + 74097065727976 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 972532690992940 T + \)\(22\!\cdots\!40\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!98\)\( T^{4} - \)\(26\!\cdots\!00\)\( p^{17} T^{5} + \)\(22\!\cdots\!40\)\( p^{34} T^{6} - 972532690992940 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 4036496333495712 T + \)\(12\!\cdots\!36\)\( T^{2} - \)\(25\!\cdots\!36\)\( T^{3} + \)\(44\!\cdots\!30\)\( T^{4} - \)\(25\!\cdots\!36\)\( p^{17} T^{5} + \)\(12\!\cdots\!36\)\( p^{34} T^{6} - 4036496333495712 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 3219563358634340 T + \)\(39\!\cdots\!72\)\( T^{2} - \)\(97\!\cdots\!88\)\( T^{3} + \)\(64\!\cdots\!02\)\( T^{4} - \)\(97\!\cdots\!88\)\( p^{17} T^{5} + \)\(39\!\cdots\!72\)\( p^{34} T^{6} - 3219563358634340 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 5269779141375740 T + \)\(30\!\cdots\!64\)\( p T^{2} - \)\(13\!\cdots\!00\)\( T^{3} - \)\(29\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!00\)\( p^{17} T^{5} + \)\(30\!\cdots\!64\)\( p^{35} T^{6} + 5269779141375740 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 1630484376609944 T + \)\(10\!\cdots\!32\)\( T^{2} + \)\(17\!\cdots\!12\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} + \)\(17\!\cdots\!12\)\( p^{17} T^{5} + \)\(10\!\cdots\!32\)\( p^{34} T^{6} - 1630484376609944 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 27965205092022016 T + \)\(98\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!36\)\( T^{3} + \)\(29\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!36\)\( p^{17} T^{5} + \)\(98\!\cdots\!04\)\( p^{34} T^{6} + 27965205092022016 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 8598163018504788 T + \)\(82\!\cdots\!64\)\( T^{2} - \)\(11\!\cdots\!80\)\( p T^{3} + \)\(36\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!80\)\( p^{18} T^{5} + \)\(82\!\cdots\!64\)\( p^{34} T^{6} - 8598163018504788 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 56353749364046152 T + \)\(48\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(98\!\cdots\!42\)\( T^{4} + \)\(21\!\cdots\!64\)\( p^{17} T^{5} + \)\(48\!\cdots\!28\)\( p^{34} T^{6} + 56353749364046152 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 360024045001148952 T + \)\(72\!\cdots\!04\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(85\!\cdots\!66\)\( T^{4} + \)\(92\!\cdots\!80\)\( p^{17} T^{5} + \)\(72\!\cdots\!04\)\( p^{34} T^{6} + 360024045001148952 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−8.006270246417175506385601225661, −7.27591239225939153931914718435, −7.07882118133159506339141223949, −6.72543112910362519449719626889, −6.68986496564999946366267610369, −6.44109850444630962387070196467, −6.07672709046295664898400970847, −5.88061114278294835491138897216, −5.87563677166270724021302378104, −5.16219519235321073974994568592, −5.03016764148748170659469145112, −5.02220901780367351494780142685, −4.72069985818814542009481081827, −4.03584445510162926015734381872, −3.91295107734186277938507318893, −3.85902941023784176543519050863, −3.79831277708240309800333456943, −2.96483741523772808335851741435, −2.86939259637803832637280052029, −2.48094742538132510556811301730, −2.41624420508351634330663627809, −1.57244425180664319407126458367, −1.46587906841559866695266285160, −1.17630245524367292947416669618, −1.08849201632035451537705276014, 0, 0, 0, 0, 1.08849201632035451537705276014, 1.17630245524367292947416669618, 1.46587906841559866695266285160, 1.57244425180664319407126458367, 2.41624420508351634330663627809, 2.48094742538132510556811301730, 2.86939259637803832637280052029, 2.96483741523772808335851741435, 3.79831277708240309800333456943, 3.85902941023784176543519050863, 3.91295107734186277938507318893, 4.03584445510162926015734381872, 4.72069985818814542009481081827, 5.02220901780367351494780142685, 5.03016764148748170659469145112, 5.16219519235321073974994568592, 5.87563677166270724021302378104, 5.88061114278294835491138897216, 6.07672709046295664898400970847, 6.44109850444630962387070196467, 6.68986496564999946366267610369, 6.72543112910362519449719626889, 7.07882118133159506339141223949, 7.27591239225939153931914718435, 8.006270246417175506385601225661