Dirichlet series
| L(s) = 1 | + 33·2-s + 163·3-s + 235·4-s + 5.37e3·6-s + 1.12e4·7-s + 5.32e3·8-s − 4.10e4·9-s − 4.01e4·11-s + 3.83e4·12-s + 1.14e5·13-s + 3.70e5·14-s + 2.42e5·16-s − 7.87e4·17-s − 1.35e6·18-s + 2.09e5·19-s + 1.83e6·21-s − 1.32e6·22-s + 4.25e6·23-s + 8.67e5·24-s + 3.77e6·26-s − 8.51e6·27-s + 2.64e6·28-s − 1.64e6·29-s − 1.13e7·31-s + 3.48e6·32-s − 6.54e6·33-s − 2.59e6·34-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.16·3-s + 0.458·4-s + 1.69·6-s + 1.76·7-s + 0.459·8-s − 2.08·9-s − 0.826·11-s + 0.533·12-s + 1.10·13-s + 2.58·14-s + 0.925·16-s − 0.228·17-s − 3.04·18-s + 0.369·19-s + 2.05·21-s − 1.20·22-s + 3.17·23-s + 0.534·24-s + 1.61·26-s − 3.08·27-s + 0.812·28-s − 0.432·29-s − 2.21·31-s + 0.587·32-s − 0.960·33-s − 0.333·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(5^{8} \cdot 13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.85024\times 10^{8}\) |
| Root analytic conductor: | \(12.9377\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 5^{8} \cdot 13^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(21.38599802\) |
| \(L(\frac12)\) | \(\approx\) | \(21.38599802\) |
| \(L(\frac{11}{2})\) | 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 | 5 | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) | |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 33 T + 427 p T^{2} - 3219 p^{3} T^{3} + 9097 p^{6} T^{4} - 3219 p^{12} T^{5} + 427 p^{19} T^{6} - 33 p^{27} T^{7} + p^{36} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 - 163 T + 67627 T^{2} - 3067592 p T^{3} + 212179120 p^{2} T^{4} - 3067592 p^{10} T^{5} + 67627 p^{18} T^{6} - 163 p^{27} T^{7} + p^{36} T^{8} \) | |
| 7 | $C_2 \wr S_4$ | \( 1 - 11241 T + 18891399 p T^{2} - 2792105460 p^{3} T^{3} + 22711650939506 p^{3} T^{4} - 2792105460 p^{12} T^{5} + 18891399 p^{19} T^{6} - 11241 p^{27} T^{7} + p^{36} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 40140 T + 5207212188 T^{2} + 260939722519836 T^{3} + 16186230268309124390 T^{4} + 260939722519836 p^{9} T^{5} + 5207212188 p^{18} T^{6} + 40140 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 78717 T + 179543541353 T^{2} - 34209748146126354 T^{3} + \)\(16\!\cdots\!82\)\( T^{4} - 34209748146126354 p^{9} T^{5} + 179543541353 p^{18} T^{6} + 78717 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 209664 T + 58556387148 p T^{2} - 9405441041308416 p T^{3} + \)\(51\!\cdots\!22\)\( T^{4} - 9405441041308416 p^{10} T^{5} + 58556387148 p^{19} T^{6} - 209664 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 4257444 T + 7366209360108 T^{2} - 5287090655000044788 T^{3} + \)\(22\!\cdots\!34\)\( T^{4} - 5287090655000044788 p^{9} T^{5} + 7366209360108 p^{18} T^{6} - 4257444 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 1647936 T + 37315906740828 T^{2} + 63599271454922849280 T^{3} + \)\(66\!\cdots\!74\)\( T^{4} + 63599271454922849280 p^{9} T^{5} + 37315906740828 p^{18} T^{6} + 1647936 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 11366002 T + 118630502542644 T^{2} + \)\(68\!\cdots\!18\)\( T^{3} + \)\(42\!\cdots\!06\)\( T^{4} + \)\(68\!\cdots\!18\)\( p^{9} T^{5} + 118630502542644 p^{18} T^{6} + 11366002 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 4636891 T + 361056879801945 T^{2} + \)\(14\!\cdots\!94\)\( T^{3} + \)\(66\!\cdots\!34\)\( T^{4} + \)\(14\!\cdots\!94\)\( p^{9} T^{5} + 361056879801945 p^{18} T^{6} + 4636891 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 13859538 T + 1246904510238356 T^{2} - \)\(13\!\cdots\!86\)\( T^{3} + \)\(60\!\cdots\!42\)\( T^{4} - \)\(13\!\cdots\!86\)\( p^{9} T^{5} + 1246904510238356 p^{18} T^{6} - 13859538 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 33368081 T + 1484927456162487 T^{2} - \)\(42\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!24\)\( T^{4} - \)\(42\!\cdots\!12\)\( p^{9} T^{5} + 1484927456162487 p^{18} T^{6} - 33368081 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 3943005 T + 2343750668926273 T^{2} - \)\(81\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!54\)\( T^{4} - \)\(81\!\cdots\!96\)\( p^{9} T^{5} + 2343750668926273 p^{18} T^{6} - 3943005 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 171019326 T + 20398534023554412 T^{2} - \)\(16\!\cdots\!86\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(16\!\cdots\!86\)\( p^{9} T^{5} + 20398534023554412 p^{18} T^{6} - 171019326 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 63389388 T + 18604706307570428 T^{2} + \)\(24\!\cdots\!68\)\( T^{3} + \)\(14\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!68\)\( p^{9} T^{5} + 18604706307570428 p^{18} T^{6} + 63389388 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 77050190 T + 39401474083082580 T^{2} - \)\(26\!\cdots\!86\)\( T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(26\!\cdots\!86\)\( p^{9} T^{5} + 39401474083082580 p^{18} T^{6} - 77050190 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 41174072 T + 67202484458082900 T^{2} - \)\(19\!\cdots\!28\)\( T^{3} + \)\(23\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!28\)\( p^{9} T^{5} + 67202484458082900 p^{18} T^{6} - 41174072 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 252460989 T + 144403365822266409 T^{2} - \)\(44\!\cdots\!68\)\( p T^{3} + \)\(90\!\cdots\!90\)\( T^{4} - \)\(44\!\cdots\!68\)\( p^{10} T^{5} + 144403365822266409 p^{18} T^{6} - 252460989 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 594415068 T + 287796385433349156 T^{2} + \)\(92\!\cdots\!76\)\( T^{3} + \)\(26\!\cdots\!46\)\( T^{4} + \)\(92\!\cdots\!76\)\( p^{9} T^{5} + 287796385433349156 p^{18} T^{6} + 594415068 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 115998984 T + 371265024783771132 T^{2} - \)\(33\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!58\)\( T^{4} - \)\(33\!\cdots\!80\)\( p^{9} T^{5} + 371265024783771132 p^{18} T^{6} - 115998984 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 79577862 T + 6204891761421028 p T^{2} - \)\(86\!\cdots\!46\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} - \)\(86\!\cdots\!46\)\( p^{9} T^{5} + 6204891761421028 p^{19} T^{6} - 79577862 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 1152240276 T + 746897787143209636 T^{2} + \)\(27\!\cdots\!00\)\( T^{3} - \)\(70\!\cdots\!14\)\( T^{4} + \)\(27\!\cdots\!00\)\( p^{9} T^{5} + 746897787143209636 p^{18} T^{6} + 1152240276 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1049098084 T + 2374666548574190388 T^{2} + \)\(23\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!74\)\( T^{4} + \)\(23\!\cdots\!60\)\( p^{9} T^{5} + 2374666548574190388 p^{18} T^{6} + 1049098084 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−6.84480316717184622055417133640, −6.82808637634212065538446234356, −5.94474604521306612240537998748, −5.91833101341503571999116386654, −5.75028776172735047247883440437, −5.33902391499462713914534215967, −5.26790420463126169116512957169, −5.07591426645797221744761090433, −4.94854705104040851643764474465, −4.51330760652882265894542930616, −4.14325285334334050457999853337, −3.97326565718226871585025494842, −3.85393382378889759551525688980, −3.31673406467772638143277696808, −3.07673616942856994122375490456, −3.05481687386237152873191995160, −2.59400727779380644495301792550, −2.45803743296687107669940525074, −2.18848374653434841095859679537, −1.65470632560460030243148934990, −1.60956098071130131317934380296, −1.11003654133433480344607577038, −0.912778303395833688043970634833, −0.52176922792296559128903886984, −0.26419437087887885870850035392, 0.26419437087887885870850035392, 0.52176922792296559128903886984, 0.912778303395833688043970634833, 1.11003654133433480344607577038, 1.60956098071130131317934380296, 1.65470632560460030243148934990, 2.18848374653434841095859679537, 2.45803743296687107669940525074, 2.59400727779380644495301792550, 3.05481687386237152873191995160, 3.07673616942856994122375490456, 3.31673406467772638143277696808, 3.85393382378889759551525688980, 3.97326565718226871585025494842, 4.14325285334334050457999853337, 4.51330760652882265894542930616, 4.94854705104040851643764474465, 5.07591426645797221744761090433, 5.26790420463126169116512957169, 5.33902391499462713914534215967, 5.75028776172735047247883440437, 5.91833101341503571999116386654, 5.94474604521306612240537998748, 6.82808637634212065538446234356, 6.84480316717184622055417133640