Dirichlet series
| L(s) = 1 | + 186·2-s − 2.78e3·3-s − 2.52e5·4-s + 2.74e5·5-s − 5.18e5·6-s − 2.30e7·7-s − 5.44e7·8-s − 2.55e8·9-s + 5.10e7·10-s + 6.10e8·11-s + 7.04e8·12-s − 8.51e9·13-s − 4.28e9·14-s − 7.65e8·15-s + 1.96e10·16-s − 4.77e10·17-s − 4.75e10·18-s − 1.42e11·19-s − 6.95e10·20-s + 6.42e10·21-s + 1.13e11·22-s + 1.61e11·23-s + 1.51e11·24-s − 5.27e11·25-s − 1.58e12·26-s + 1.51e12·27-s + 5.83e12·28-s + ⋯ |
| L(s) = 1 | + 0.513·2-s − 0.245·3-s − 1.93·4-s + 0.314·5-s − 0.125·6-s − 1.51·7-s − 1.14·8-s − 1.97·9-s + 0.161·10-s + 0.858·11-s + 0.473·12-s − 2.89·13-s − 0.776·14-s − 0.0771·15-s + 1.14·16-s − 1.66·17-s − 1.01·18-s − 1.92·19-s − 0.607·20-s + 0.370·21-s + 0.440·22-s + 0.429·23-s + 0.281·24-s − 0.691·25-s − 1.48·26-s + 1.03·27-s + 2.91·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2401\) = \(7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(27058.4\) |
| Root analytic conductor: | \(3.58127\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2401,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
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 | 7 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 93 p T + 35949 p^{3} T^{2} - 180093 p^{8} T^{3} + 25177105 p^{11} T^{4} - 180093 p^{25} T^{5} + 35949 p^{37} T^{6} - 93 p^{52} T^{7} + p^{68} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 2786 T + 9747656 p^{3} T^{2} - 306459314 p^{5} T^{3} + 48872111771182 p^{6} T^{4} - 306459314 p^{22} T^{5} + 9747656 p^{37} T^{6} + 2786 p^{51} T^{7} + p^{68} T^{8} \) | |
| 5 | $C_2 \wr S_4$ | \( 1 - 274722 T + 602669145396 T^{2} + 2886269096223786 p^{3} T^{3} + \)\(63\!\cdots\!74\)\( p^{4} T^{4} + 2886269096223786 p^{20} T^{5} + 602669145396 p^{34} T^{6} - 274722 p^{51} T^{7} + p^{68} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 610110180 T + 7005149347042788 p T^{2} - \)\(14\!\cdots\!48\)\( p^{2} T^{3} + \)\(23\!\cdots\!58\)\( p^{3} T^{4} - \)\(14\!\cdots\!48\)\( p^{19} T^{5} + 7005149347042788 p^{35} T^{6} - 610110180 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 8514921674 T + 45947279477181818284 T^{2} + \)\(13\!\cdots\!18\)\( p T^{3} + \)\(35\!\cdots\!54\)\( p^{2} T^{4} + \)\(13\!\cdots\!18\)\( p^{18} T^{5} + 45947279477181818284 p^{34} T^{6} + 8514921674 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 47762899716 T + \)\(27\!\cdots\!64\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(33\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!08\)\( p^{17} T^{5} + \)\(27\!\cdots\!64\)\( p^{34} T^{6} + 47762899716 p^{51} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 142813479494 T + \)\(20\!\cdots\!36\)\( T^{2} + \)\(18\!\cdots\!34\)\( T^{3} + \)\(17\!\cdots\!86\)\( T^{4} + \)\(18\!\cdots\!34\)\( p^{17} T^{5} + \)\(20\!\cdots\!36\)\( p^{34} T^{6} + 142813479494 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 161322432240 T + \)\(35\!\cdots\!48\)\( T^{2} - \)\(91\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!14\)\( T^{4} - \)\(91\!\cdots\!76\)\( p^{17} T^{5} + \)\(35\!\cdots\!48\)\( p^{34} T^{6} - 161322432240 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 2470023989364 T + \)\(28\!\cdots\!56\)\( T^{2} - \)\(52\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!86\)\( T^{4} - \)\(52\!\cdots\!64\)\( p^{17} T^{5} + \)\(28\!\cdots\!56\)\( p^{34} T^{6} - 2470023989364 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 3069063677988 T + \)\(62\!\cdots\!56\)\( T^{2} - \)\(23\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!90\)\( T^{4} - \)\(23\!\cdots\!44\)\( p^{17} T^{5} + \)\(62\!\cdots\!56\)\( p^{34} T^{6} - 3069063677988 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 53477713304508 T + \)\(21\!\cdots\!40\)\( T^{2} + \)\(64\!\cdots\!56\)\( T^{3} + \)\(15\!\cdots\!78\)\( T^{4} + \)\(64\!\cdots\!56\)\( p^{17} T^{5} + \)\(21\!\cdots\!40\)\( p^{34} T^{6} + 53477713304508 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 84856086719628 T + \)\(40\!\cdots\!44\)\( T^{2} - \)\(13\!\cdots\!28\)\( T^{3} - \)\(11\!\cdots\!90\)\( T^{4} - \)\(13\!\cdots\!28\)\( p^{17} T^{5} + \)\(40\!\cdots\!44\)\( p^{34} T^{6} + 84856086719628 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 14664094189676 T + \)\(15\!\cdots\!20\)\( T^{2} - \)\(27\!\cdots\!52\)\( T^{3} + \)\(12\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!52\)\( p^{17} T^{5} + \)\(15\!\cdots\!20\)\( p^{34} T^{6} - 14664094189676 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 110590112906028 T + \)\(17\!\cdots\!64\)\( p T^{2} - \)\(77\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!50\)\( T^{4} - \)\(77\!\cdots\!72\)\( p^{17} T^{5} + \)\(17\!\cdots\!64\)\( p^{35} T^{6} - 110590112906028 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 517697020820328 T + \)\(41\!\cdots\!96\)\( T^{2} + \)\(22\!\cdots\!72\)\( T^{3} + \)\(94\!\cdots\!70\)\( T^{4} + \)\(22\!\cdots\!72\)\( p^{17} T^{5} + \)\(41\!\cdots\!96\)\( p^{34} T^{6} + 517697020820328 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 2778221577040518 T + \)\(71\!\cdots\!92\)\( T^{2} + \)\(10\!\cdots\!66\)\( T^{3} + \)\(14\!\cdots\!74\)\( T^{4} + \)\(10\!\cdots\!66\)\( p^{17} T^{5} + \)\(71\!\cdots\!92\)\( p^{34} T^{6} + 2778221577040518 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 3329268765288494 T + \)\(87\!\cdots\!96\)\( T^{2} + \)\(17\!\cdots\!66\)\( T^{3} + \)\(30\!\cdots\!46\)\( T^{4} + \)\(17\!\cdots\!66\)\( p^{17} T^{5} + \)\(87\!\cdots\!96\)\( p^{34} T^{6} + 3329268765288494 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 1542501737022560 T + \)\(33\!\cdots\!56\)\( T^{2} + \)\(24\!\cdots\!92\)\( T^{3} + \)\(48\!\cdots\!94\)\( T^{4} + \)\(24\!\cdots\!92\)\( p^{17} T^{5} + \)\(33\!\cdots\!56\)\( p^{34} T^{6} + 1542501737022560 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 739192660163448 T + \)\(69\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(22\!\cdots\!58\)\( T^{4} + \)\(11\!\cdots\!08\)\( p^{17} T^{5} + \)\(69\!\cdots\!80\)\( p^{34} T^{6} - 739192660163448 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1768956758306304 T + \)\(15\!\cdots\!96\)\( T^{2} + \)\(34\!\cdots\!28\)\( T^{3} + \)\(98\!\cdots\!82\)\( T^{4} + \)\(34\!\cdots\!28\)\( p^{17} T^{5} + \)\(15\!\cdots\!96\)\( p^{34} T^{6} + 1768956758306304 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1824334397417480 T + \)\(42\!\cdots\!76\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - \)\(11\!\cdots\!24\)\( p^{17} T^{5} + \)\(42\!\cdots\!76\)\( p^{34} T^{6} - 1824334397417480 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 49183281603250158 T + \)\(21\!\cdots\!68\)\( T^{2} + \)\(54\!\cdots\!54\)\( T^{3} + \)\(13\!\cdots\!78\)\( T^{4} + \)\(54\!\cdots\!54\)\( p^{17} T^{5} + \)\(21\!\cdots\!68\)\( p^{34} T^{6} + 49183281603250158 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 138655399526807016 T + \)\(12\!\cdots\!76\)\( T^{2} + \)\(69\!\cdots\!16\)\( T^{3} + \)\(30\!\cdots\!86\)\( T^{4} + \)\(69\!\cdots\!16\)\( p^{17} T^{5} + \)\(12\!\cdots\!76\)\( p^{34} T^{6} + 138655399526807016 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 73446338844858924 T + \)\(14\!\cdots\!12\)\( T^{2} - \)\(83\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(83\!\cdots\!08\)\( p^{17} T^{5} + \)\(14\!\cdots\!12\)\( p^{34} T^{6} - 73446338844858924 p^{51} T^{7} + p^{68} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.70413305399304772956301430618, −12.94660789391145369319820301117, −12.85646130593098064869293145469, −12.30171392840691537539756118513, −12.09112476852973258962726564494, −11.57683756330827822323843399193, −10.89531739156281558371572629807, −10.23497381352073843148526902452, −10.07041320197733143846964726099, −9.389885379481557048368265100928, −9.002782088425977706109409639626, −8.781824027192336686321572558517, −8.609617000321132520792982421120, −7.63888514420479483316762093094, −6.81346863017482316365978394549, −6.50765577649262657274166895687, −6.14295130813690473265562724757, −5.25566293256639367013452505876, −5.04016220444712589454411778047, −4.34561744255245332913797919953, −4.32222620163732425833173500340, −3.32260993172969543325913931935, −2.75452067148864792035006208988, −2.48611652886379038073505392355, −1.59257517371165018500067054973, 0, 0, 0, 0, 1.59257517371165018500067054973, 2.48611652886379038073505392355, 2.75452067148864792035006208988, 3.32260993172969543325913931935, 4.32222620163732425833173500340, 4.34561744255245332913797919953, 5.04016220444712589454411778047, 5.25566293256639367013452505876, 6.14295130813690473265562724757, 6.50765577649262657274166895687, 6.81346863017482316365978394549, 7.63888514420479483316762093094, 8.609617000321132520792982421120, 8.781824027192336686321572558517, 9.002782088425977706109409639626, 9.389885379481557048368265100928, 10.07041320197733143846964726099, 10.23497381352073843148526902452, 10.89531739156281558371572629807, 11.57683756330827822323843399193, 12.09112476852973258962726564494, 12.30171392840691537539756118513, 12.85646130593098064869293145469, 12.94660789391145369319820301117, 13.70413305399304772956301430618