Dirichlet series
| L(s) = 1 | − 33·2-s − 163·3-s + 235·4-s + 471·5-s + 5.37e3·6-s − 1.12e4·7-s − 5.32e3·8-s − 4.10e4·9-s − 1.55e4·10-s − 4.01e4·11-s − 3.83e4·12-s − 1.14e5·13-s + 3.70e5·14-s − 7.67e4·15-s + 2.42e5·16-s + 7.87e4·17-s + 1.35e6·18-s + 2.09e5·19-s + 1.10e5·20-s + 1.83e6·21-s + 1.32e6·22-s − 4.25e6·23-s + 8.67e5·24-s − 5.24e6·25-s + 3.77e6·26-s + 8.51e6·27-s − 2.64e6·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 1.16·3-s + 0.458·4-s + 0.337·5-s + 1.69·6-s − 1.76·7-s − 0.459·8-s − 2.08·9-s − 0.491·10-s − 0.826·11-s − 0.533·12-s − 1.10·13-s + 2.58·14-s − 0.391·15-s + 0.925·16-s + 0.228·17-s + 3.04·18-s + 0.369·19-s + 0.154·20-s + 2.05·21-s + 1.20·22-s − 3.17·23-s + 0.534·24-s − 2.68·25-s + 1.61·26-s + 3.08·27-s − 0.812·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(28561\) = \(13^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2009.66\) |
| Root analytic conductor: | \(2.58755\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 28561,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | 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 | 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 | $C_2 \wr S_4$ | \( 1 + 163 T + 67627 T^{2} + 3067592 p T^{3} + 212179120 p^{2} T^{4} + 3067592 p^{10} T^{5} + 67627 p^{18} T^{6} + 163 p^{27} T^{7} + p^{36} T^{8} \) | |
| 5 | $C_2 \wr S_4$ | \( 1 - 471 T + 5467249 T^{2} - 2978826294 T^{3} + 2762975487438 p T^{4} - 2978826294 p^{9} T^{5} + 5467249 p^{18} T^{6} - 471 p^{27} T^{7} + p^{36} T^{8} \) | |
| 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
−13.95928460430890317870092855744, −12.97106901514597117453914352925, −12.85445491453793972570809894225, −12.43626443260267289556638154263, −11.83918470565123820424209471048, −11.75124371675164354200421837086, −11.31969655516604555701088514367, −10.94595480247553116440448571098, −10.26495302653789196245989292064, −9.892948414715274431333698709278, −9.608986960805973076664105819727, −9.274698042266345077491294788431, −9.196526681358649258536784772497, −8.044205686931871745573286906803, −8.039987945857310034350189533328, −7.82905030775083749862309007042, −6.57639634467340707158950518251, −6.31081068795648558770452480345, −5.84923773055124293501359703709, −5.54522832179349602028947311014, −5.19631160721890757832532363900, −3.73592644965051715793501526344, −3.38532871414856603989847282361, −2.51114059648194216727569569543, −1.97447730592761554791926188116, 0, 0, 0, 0, 1.97447730592761554791926188116, 2.51114059648194216727569569543, 3.38532871414856603989847282361, 3.73592644965051715793501526344, 5.19631160721890757832532363900, 5.54522832179349602028947311014, 5.84923773055124293501359703709, 6.31081068795648558770452480345, 6.57639634467340707158950518251, 7.82905030775083749862309007042, 8.039987945857310034350189533328, 8.044205686931871745573286906803, 9.196526681358649258536784772497, 9.274698042266345077491294788431, 9.608986960805973076664105819727, 9.892948414715274431333698709278, 10.26495302653789196245989292064, 10.94595480247553116440448571098, 11.31969655516604555701088514367, 11.75124371675164354200421837086, 11.83918470565123820424209471048, 12.43626443260267289556638154263, 12.85445491453793972570809894225, 12.97106901514597117453914352925, 13.95928460430890317870092855744