Dirichlet series
| L(s) = 1 | − 40·2-s + 28·3-s + 960·4-s + 152·5-s − 1.12e3·6-s − 132·7-s − 1.79e4·8-s − 501·9-s − 6.08e3·10-s − 4.21e3·11-s + 2.68e4·12-s − 7.01e3·13-s + 5.28e3·14-s + 4.25e3·15-s + 2.86e5·16-s − 3.01e4·17-s + 2.00e4·18-s − 8.32e3·19-s + 1.45e5·20-s − 3.69e3·21-s + 1.68e5·22-s + 6.71e4·23-s − 5.01e5·24-s + 3.69e3·25-s + 2.80e5·26-s + 1.32e5·27-s − 1.26e5·28-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 0.598·3-s + 15/2·4-s + 0.543·5-s − 2.11·6-s − 0.145·7-s − 12.3·8-s − 0.229·9-s − 1.92·10-s − 0.955·11-s + 4.49·12-s − 0.885·13-s + 0.514·14-s + 0.325·15-s + 35/2·16-s − 1.48·17-s + 0.809·18-s − 0.278·19-s + 4.07·20-s − 0.0870·21-s + 3.37·22-s + 1.15·23-s − 7.40·24-s + 0.0472·25-s + 3.13·26-s + 1.29·27-s − 1.09·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(10\) |
| Conductor: | \(2^{5} \cdot 31^{5}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.72526\times 10^{6}\) |
| Root analytic conductor: | \(4.40089\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((10,\ 2^{5} \cdot 31^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.01540106653\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01540106653\) |
| \(L(\frac{9}{2})\) | 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 | $C_1$ | \( ( 1 + p^{3} T )^{5} \) |
| 31 | $C_1$ | \( ( 1 + p^{3} T )^{5} \) | |
| good | 3 | $C_2 \wr S_5$ | \( 1 - 28 T + 1285 T^{2} - 60812 p T^{3} + 3008944 p T^{4} - 10986400 p^{2} T^{5} + 3008944 p^{8} T^{6} - 60812 p^{15} T^{7} + 1285 p^{21} T^{8} - 28 p^{28} T^{9} + p^{35} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 152 T + 3882 p T^{2} + 1002186 p^{2} T^{3} + 67228441 p^{3} T^{4} - 3594590756 p^{4} T^{5} + 67228441 p^{10} T^{6} + 1002186 p^{16} T^{7} + 3882 p^{22} T^{8} - 152 p^{28} T^{9} + p^{35} T^{10} \) | |
| 7 | $C_2 \wr S_5$ | \( 1 + 132 T + 2058114 T^{2} - 8356172 T^{3} + 2580262843201 T^{4} + 108126606206672 T^{5} + 2580262843201 p^{7} T^{6} - 8356172 p^{14} T^{7} + 2058114 p^{21} T^{8} + 132 p^{28} T^{9} + p^{35} T^{10} \) | |
| 11 | $C_2 \wr S_5$ | \( 1 + 4218 T + 3613987 p T^{2} + 151723008280 T^{3} + 889140040292404 T^{4} + 3392376385870577404 T^{5} + 889140040292404 p^{7} T^{6} + 151723008280 p^{14} T^{7} + 3613987 p^{22} T^{8} + 4218 p^{28} T^{9} + p^{35} T^{10} \) | |
| 13 | $C_2 \wr S_5$ | \( 1 + 7014 T + 157643847 T^{2} + 277858864264 T^{3} + 7729043099542108 T^{4} - 21080940566200026156 T^{5} + 7729043099542108 p^{7} T^{6} + 277858864264 p^{14} T^{7} + 157643847 p^{21} T^{8} + 7014 p^{28} T^{9} + p^{35} T^{10} \) | |
| 17 | $C_2 \wr S_5$ | \( 1 + 30180 T + 914386897 T^{2} + 22554992591576 T^{3} + 571851490828371838 T^{4} + \)\(98\!\cdots\!80\)\( T^{5} + 571851490828371838 p^{7} T^{6} + 22554992591576 p^{14} T^{7} + 914386897 p^{21} T^{8} + 30180 p^{28} T^{9} + p^{35} T^{10} \) | |
| 19 | $C_2 \wr S_5$ | \( 1 + 8320 T + 1803066718 T^{2} + 415523156220 p T^{3} + 2300299973249642769 T^{4} + \)\(10\!\cdots\!68\)\( T^{5} + 2300299973249642769 p^{7} T^{6} + 415523156220 p^{15} T^{7} + 1803066718 p^{21} T^{8} + 8320 p^{28} T^{9} + p^{35} T^{10} \) | |
| 23 | $C_2 \wr S_5$ | \( 1 - 67130 T + 13577754263 T^{2} - 907010958732408 T^{3} + 81897282425535118862 T^{4} - \)\(46\!\cdots\!56\)\( T^{5} + 81897282425535118862 p^{7} T^{6} - 907010958732408 p^{14} T^{7} + 13577754263 p^{21} T^{8} - 67130 p^{28} T^{9} + p^{35} T^{10} \) | |
| 29 | $C_2 \wr S_5$ | \( 1 - 290034 T + 67640775935 T^{2} - 11475606060990608 T^{3} + \)\(20\!\cdots\!32\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!32\)\( p^{7} T^{6} - 11475606060990608 p^{14} T^{7} + 67640775935 p^{21} T^{8} - 290034 p^{28} T^{9} + p^{35} T^{10} \) | |
| 37 | $C_2 \wr S_5$ | \( 1 - 682400 T + 640954707751 T^{2} - 277367275618873828 T^{3} + \)\(13\!\cdots\!48\)\( T^{4} - \)\(40\!\cdots\!88\)\( T^{5} + \)\(13\!\cdots\!48\)\( p^{7} T^{6} - 277367275618873828 p^{14} T^{7} + 640954707751 p^{21} T^{8} - 682400 p^{28} T^{9} + p^{35} T^{10} \) | |
| 41 | $C_2 \wr S_5$ | \( 1 - 197844 T + 403930801826 T^{2} - 107052006322873654 T^{3} + \)\(10\!\cdots\!97\)\( T^{4} - \)\(87\!\cdots\!72\)\( p T^{5} + \)\(10\!\cdots\!97\)\( p^{7} T^{6} - 107052006322873654 p^{14} T^{7} + 403930801826 p^{21} T^{8} - 197844 p^{28} T^{9} + p^{35} T^{10} \) | |
| 43 | $C_2 \wr S_5$ | \( 1 - 2333288 T + 3015419382805 T^{2} - 2659962875132848612 T^{3} + \)\(18\!\cdots\!80\)\( T^{4} - \)\(10\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!80\)\( p^{7} T^{6} - 2659962875132848612 p^{14} T^{7} + 3015419382805 p^{21} T^{8} - 2333288 p^{28} T^{9} + p^{35} T^{10} \) | |
| 47 | $C_2 \wr S_5$ | \( 1 + 405160 T + 1639147055183 T^{2} + 971233682632407856 T^{3} + \)\(12\!\cdots\!06\)\( T^{4} + \)\(77\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!06\)\( p^{7} T^{6} + 971233682632407856 p^{14} T^{7} + 1639147055183 p^{21} T^{8} + 405160 p^{28} T^{9} + p^{35} T^{10} \) | |
| 53 | $C_2 \wr S_5$ | \( 1 - 1029596 T + 3195449895635 T^{2} - 1032746699078462644 T^{3} + \)\(33\!\cdots\!52\)\( T^{4} + \)\(40\!\cdots\!76\)\( T^{5} + \)\(33\!\cdots\!52\)\( p^{7} T^{6} - 1032746699078462644 p^{14} T^{7} + 3195449895635 p^{21} T^{8} - 1029596 p^{28} T^{9} + p^{35} T^{10} \) | |
| 59 | $C_2 \wr S_5$ | \( 1 + 400004 T + 2747533619654 T^{2} - 3512908734231126336 T^{3} + \)\(13\!\cdots\!65\)\( T^{4} + \)\(35\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!65\)\( p^{7} T^{6} - 3512908734231126336 p^{14} T^{7} + 2747533619654 p^{21} T^{8} + 400004 p^{28} T^{9} + p^{35} T^{10} \) | |
| 61 | $C_2 \wr S_5$ | \( 1 + 4933754 T + 16710701531271 T^{2} + 35136076191349087264 T^{3} + \)\(67\!\cdots\!72\)\( T^{4} + \)\(11\!\cdots\!24\)\( T^{5} + \)\(67\!\cdots\!72\)\( p^{7} T^{6} + 35136076191349087264 p^{14} T^{7} + 16710701531271 p^{21} T^{8} + 4933754 p^{28} T^{9} + p^{35} T^{10} \) | |
| 67 | $C_2 \wr S_5$ | \( 1 - 2699572 T + 9945589528023 T^{2} - 10389941561616657904 T^{3} + \)\(72\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!28\)\( T^{5} + \)\(72\!\cdots\!90\)\( p^{7} T^{6} - 10389941561616657904 p^{14} T^{7} + 9945589528023 p^{21} T^{8} - 2699572 p^{28} T^{9} + p^{35} T^{10} \) | |
| 71 | $C_2 \wr S_5$ | \( 1 + 613944 T + 36439660785006 T^{2} + 5681449290686073080 T^{3} + \)\(57\!\cdots\!97\)\( T^{4} + \)\(41\!\cdots\!80\)\( T^{5} + \)\(57\!\cdots\!97\)\( p^{7} T^{6} + 5681449290686073080 p^{14} T^{7} + 36439660785006 p^{21} T^{8} + 613944 p^{28} T^{9} + p^{35} T^{10} \) | |
| 73 | $C_2 \wr S_5$ | \( 1 - 7894998 T + 55779407414437 T^{2} - \)\(26\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!30\)\( T^{4} - \)\(39\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!30\)\( p^{7} T^{6} - \)\(26\!\cdots\!72\)\( p^{14} T^{7} + 55779407414437 p^{21} T^{8} - 7894998 p^{28} T^{9} + p^{35} T^{10} \) | |
| 79 | $C_2 \wr S_5$ | \( 1 - 10463774 T + 122211092974383 T^{2} - \)\(79\!\cdots\!56\)\( T^{3} + \)\(51\!\cdots\!62\)\( T^{4} - \)\(22\!\cdots\!80\)\( T^{5} + \)\(51\!\cdots\!62\)\( p^{7} T^{6} - \)\(79\!\cdots\!56\)\( p^{14} T^{7} + 122211092974383 p^{21} T^{8} - 10463774 p^{28} T^{9} + p^{35} T^{10} \) | |
| 83 | $C_2 \wr S_5$ | \( 1 + 12217942 T + 149558627433633 T^{2} + \)\(12\!\cdots\!76\)\( T^{3} + \)\(82\!\cdots\!76\)\( T^{4} + \)\(47\!\cdots\!48\)\( T^{5} + \)\(82\!\cdots\!76\)\( p^{7} T^{6} + \)\(12\!\cdots\!76\)\( p^{14} T^{7} + 149558627433633 p^{21} T^{8} + 12217942 p^{28} T^{9} + p^{35} T^{10} \) | |
| 89 | $C_2 \wr S_5$ | \( 1 - 22649884 T + 382150584614097 T^{2} - \)\(42\!\cdots\!68\)\( T^{3} + \)\(40\!\cdots\!82\)\( T^{4} - \)\(28\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!82\)\( p^{7} T^{6} - \)\(42\!\cdots\!68\)\( p^{14} T^{7} + 382150584614097 p^{21} T^{8} - 22649884 p^{28} T^{9} + p^{35} T^{10} \) | |
| 97 | $C_2 \wr S_5$ | \( 1 + 7610048 T + 139608980995882 T^{2} + \)\(13\!\cdots\!34\)\( T^{3} + \)\(12\!\cdots\!01\)\( T^{4} + \)\(13\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!01\)\( p^{7} T^{6} + \)\(13\!\cdots\!34\)\( p^{14} T^{7} + 139608980995882 p^{21} T^{8} + 7610048 p^{28} T^{9} + p^{35} T^{10} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.053009029877443457917126173521, −7.87978996895800783471816360756, −7.70519334054524847901291400947, −7.63277654859942154501364671890, −7.18742813436333861158256142032, −6.83764982337566587207023323332, −6.65198506457889517265242295334, −6.38566289369355687387499095341, −6.08292093735977818963557547250, −6.06806337251435702015947357570, −5.33177669503366375649278210753, −5.08846325504703495801123251332, −4.74416666258073524725279468891, −4.32945022398659935633408604638, −3.87307327377466591970747836605, −3.15777117520131615378624195888, −2.86864830486608979768701266036, −2.52008750993819193737567817866, −2.43881213751726958100766672847, −2.38384922037735843668654682209, −1.68045492747871363867772305877, −1.07289814588735869390369453757, −1.04361524976351441433578176226, −0.63887071758918474890910204771, −0.03600905635187031383214116715, 0.03600905635187031383214116715, 0.63887071758918474890910204771, 1.04361524976351441433578176226, 1.07289814588735869390369453757, 1.68045492747871363867772305877, 2.38384922037735843668654682209, 2.43881213751726958100766672847, 2.52008750993819193737567817866, 2.86864830486608979768701266036, 3.15777117520131615378624195888, 3.87307327377466591970747836605, 4.32945022398659935633408604638, 4.74416666258073524725279468891, 5.08846325504703495801123251332, 5.33177669503366375649278210753, 6.06806337251435702015947357570, 6.08292093735977818963557547250, 6.38566289369355687387499095341, 6.65198506457889517265242295334, 6.83764982337566587207023323332, 7.18742813436333861158256142032, 7.63277654859942154501364671890, 7.70519334054524847901291400947, 7.87978996895800783471816360756, 8.053009029877443457917126173521