Dirichlet series
| L(s) = 1 | − 20·3-s − 1.14e4·7-s − 1.78e5·9-s − 8.75e5·11-s + 1.51e6·13-s + 3.13e6·17-s − 7.20e6·19-s + 2.28e5·21-s − 3.33e7·23-s − 1.21e7·27-s + 3.29e7·29-s − 7.25e7·31-s + 1.75e7·33-s + 5.60e8·37-s − 3.02e7·39-s + 5.50e8·41-s − 1.13e9·43-s − 8.31e8·47-s − 4.78e9·49-s − 6.27e7·51-s + 3.87e9·53-s + 1.44e8·57-s − 2.28e9·59-s + 1.00e10·61-s + 2.03e9·63-s + 2.25e10·67-s + 6.67e8·69-s + ⋯ |
| L(s) = 1 | − 0.0475·3-s − 0.257·7-s − 1.00·9-s − 1.63·11-s + 1.13·13-s + 0.536·17-s − 0.667·19-s + 0.0122·21-s − 1.08·23-s − 0.162·27-s + 0.298·29-s − 0.454·31-s + 0.0779·33-s + 1.32·37-s − 0.0537·39-s + 0.741·41-s − 1.18·43-s − 0.528·47-s − 2.42·49-s − 0.0254·51-s + 1.27·53-s + 0.0317·57-s − 0.416·59-s + 1.52·61-s + 0.258·63-s + 2.04·67-s + 0.0513·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.48514\times 10^{7}\) |
| Root analytic conductor: | \(8.76551\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 11/2, 11/2, 11/2, 11/2 ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(3.796592342\) |
| \(L(\frac12)\) | \(\approx\) | \(3.796592342\) |
| \(L(\frac{13}{2})\) | 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 | \( 1 \) | |
| 5 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_4$ | \( 1 + 20 T + 59522 p T^{2} + 713600 p^{3} T^{3} + 48581549 p^{5} T^{4} + 713600 p^{14} T^{5} + 59522 p^{23} T^{6} + 20 p^{33} T^{7} + p^{44} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 11440 T + 4918124020 T^{2} + 14527452874960 p T^{3} + 248739493448382502 p^{2} T^{4} + 14527452874960 p^{12} T^{5} + 4918124020 p^{22} T^{6} + 11440 p^{33} T^{7} + p^{44} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 79620 p T + 606917849294 T^{2} + 19523335372392960 p T^{3} + \)\(12\!\cdots\!51\)\( T^{4} + 19523335372392960 p^{12} T^{5} + 606917849294 p^{22} T^{6} + 79620 p^{34} T^{7} + p^{44} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 1513880 T + 2292564266836 T^{2} - 541025796454913000 T^{3} + \)\(22\!\cdots\!62\)\( T^{4} - 541025796454913000 p^{11} T^{5} + 2292564266836 p^{22} T^{6} - 1513880 p^{33} T^{7} + p^{44} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 3138060 T + 102234211403234 T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(46\!\cdots\!67\)\( T^{4} - \)\(29\!\cdots\!00\)\( p^{11} T^{5} + 102234211403234 p^{22} T^{6} - 3138060 p^{33} T^{7} + p^{44} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 7202644 T + 357138096085702 T^{2} + \)\(25\!\cdots\!32\)\( T^{3} + \)\(57\!\cdots\!95\)\( T^{4} + \)\(25\!\cdots\!32\)\( p^{11} T^{5} + 357138096085702 p^{22} T^{6} + 7202644 p^{33} T^{7} + p^{44} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 33351720 T + 2572975219434020 T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!58\)\( T^{4} + \)\(91\!\cdots\!40\)\( p^{11} T^{5} + 2572975219434020 p^{22} T^{6} + 33351720 p^{33} T^{7} + p^{44} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 32996856 T + 1395930398902292 T^{2} - \)\(24\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!70\)\( T^{4} - \)\(24\!\cdots\!48\)\( p^{11} T^{5} + 1395930398902292 p^{22} T^{6} - 32996856 p^{33} T^{7} + p^{44} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 72522616 T + 46916916173495620 T^{2} + \)\(42\!\cdots\!44\)\( T^{3} + \)\(97\!\cdots\!74\)\( T^{4} + \)\(42\!\cdots\!44\)\( p^{11} T^{5} + 46916916173495620 p^{22} T^{6} + 72522616 p^{33} T^{7} + p^{44} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 560863040 T + 419276381288280220 T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(19\!\cdots\!20\)\( p^{11} T^{5} + 419276381288280220 p^{22} T^{6} - 560863040 p^{33} T^{7} + p^{44} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 550137204 T + 1126206597196652570 T^{2} - \)\(83\!\cdots\!96\)\( T^{3} + \)\(64\!\cdots\!79\)\( T^{4} - \)\(83\!\cdots\!96\)\( p^{11} T^{5} + 1126206597196652570 p^{22} T^{6} - 550137204 p^{33} T^{7} + p^{44} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 1138695160 T + 2326700528359633228 T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(56\!\cdots\!58\)\( p T^{4} + \)\(21\!\cdots\!60\)\( p^{11} T^{5} + 2326700528359633228 p^{22} T^{6} + 1138695160 p^{33} T^{7} + p^{44} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 831048600 T + 5561702139339449084 T^{2} + \)\(74\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!82\)\( T^{4} + \)\(74\!\cdots\!00\)\( p^{11} T^{5} + 5561702139339449084 p^{22} T^{6} + 831048600 p^{33} T^{7} + p^{44} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 3870049560 T + 38044411998439328780 T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!20\)\( p^{11} T^{5} + 38044411998439328780 p^{22} T^{6} - 3870049560 p^{33} T^{7} + p^{44} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 2289423768 T + 34648412359443459020 T^{2} - \)\(58\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!94\)\( T^{4} - \)\(58\!\cdots\!12\)\( p^{11} T^{5} + 34648412359443459020 p^{22} T^{6} + 2289423768 p^{33} T^{7} + p^{44} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 10056795248 T + \)\(12\!\cdots\!08\)\( T^{2} - \)\(85\!\cdots\!96\)\( T^{3} + \)\(76\!\cdots\!70\)\( T^{4} - \)\(85\!\cdots\!96\)\( p^{11} T^{5} + \)\(12\!\cdots\!08\)\( p^{22} T^{6} - 10056795248 p^{33} T^{7} + p^{44} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 22570191740 T + \)\(51\!\cdots\!50\)\( T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!03\)\( T^{4} - \)\(65\!\cdots\!20\)\( p^{11} T^{5} + \)\(51\!\cdots\!50\)\( p^{22} T^{6} - 22570191740 p^{33} T^{7} + p^{44} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 27084161592 T + \)\(74\!\cdots\!08\)\( T^{2} - \)\(92\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} - \)\(92\!\cdots\!64\)\( p^{11} T^{5} + \)\(74\!\cdots\!08\)\( p^{22} T^{6} - 27084161592 p^{33} T^{7} + p^{44} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 5474005420 T + \)\(30\!\cdots\!46\)\( T^{2} + \)\(32\!\cdots\!00\)\( T^{3} + \)\(74\!\cdots\!87\)\( T^{4} + \)\(32\!\cdots\!00\)\( p^{11} T^{5} + \)\(30\!\cdots\!46\)\( p^{22} T^{6} + 5474005420 p^{33} T^{7} + p^{44} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 74053361152 T + \)\(49\!\cdots\!80\)\( T^{2} + \)\(18\!\cdots\!12\)\( T^{3} + \)\(62\!\cdots\!94\)\( T^{4} + \)\(18\!\cdots\!12\)\( p^{11} T^{5} + \)\(49\!\cdots\!80\)\( p^{22} T^{6} + 74053361152 p^{33} T^{7} + p^{44} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 94129481100 T + \)\(74\!\cdots\!90\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!03\)\( T^{4} - \)\(35\!\cdots\!00\)\( p^{11} T^{5} + \)\(74\!\cdots\!90\)\( p^{22} T^{6} - 94129481100 p^{33} T^{7} + p^{44} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 105091864236 T + \)\(10\!\cdots\!42\)\( T^{2} + \)\(56\!\cdots\!28\)\( T^{3} + \)\(36\!\cdots\!95\)\( T^{4} + \)\(56\!\cdots\!28\)\( p^{11} T^{5} + \)\(10\!\cdots\!42\)\( p^{22} T^{6} + 105091864236 p^{33} T^{7} + p^{44} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 327217174520 T + \)\(41\!\cdots\!00\)\( T^{2} - \)\(25\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(25\!\cdots\!60\)\( p^{11} T^{5} + \)\(41\!\cdots\!00\)\( p^{22} T^{6} - 327217174520 p^{33} T^{7} + p^{44} T^{8} \) | |
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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
−8.279032001856584581082325354920, −7.80861508678237729183968799093, −7.34771176715387665095593527406, −7.17832330076305624825856739458, −6.95614684623935529406329868038, −6.19224591456021984909007662757, −6.07078490213012428120899722171, −6.04282427902360917265939646426, −5.88376787617793422539432696590, −5.19283079824965582735607451293, −4.90121227407382341508955360123, −4.88986335045459431999409330846, −4.44683944875905315791901199598, −3.80117318975481654676382209396, −3.60781561911697160042633468052, −3.41087120300672167680625660021, −3.08438271014823283138004313526, −2.62860511454326838380404452375, −2.25615455490662198154593472494, −2.02392300187257688852214633880, −1.83660841638293086874936041076, −1.15690268864713591010717019199, −0.74144462283061572041327964278, −0.46534122623313138987873325763, −0.34372399929751790842217911289, 0.34372399929751790842217911289, 0.46534122623313138987873325763, 0.74144462283061572041327964278, 1.15690268864713591010717019199, 1.83660841638293086874936041076, 2.02392300187257688852214633880, 2.25615455490662198154593472494, 2.62860511454326838380404452375, 3.08438271014823283138004313526, 3.41087120300672167680625660021, 3.60781561911697160042633468052, 3.80117318975481654676382209396, 4.44683944875905315791901199598, 4.88986335045459431999409330846, 4.90121227407382341508955360123, 5.19283079824965582735607451293, 5.88376787617793422539432696590, 6.04282427902360917265939646426, 6.07078490213012428120899722171, 6.19224591456021984909007662757, 6.95614684623935529406329868038, 7.17832330076305624825856739458, 7.34771176715387665095593527406, 7.80861508678237729183968799093, 8.279032001856584581082325354920