Dirichlet series
| L(s) = 1 | − 2.62e4·3-s + 5.28e5·5-s + 2.30e7·7-s + 4.30e8·9-s − 2.69e8·11-s + 1.98e9·13-s − 1.38e10·15-s − 2.97e10·17-s − 9.16e10·19-s − 6.05e11·21-s + 3.83e10·23-s − 1.50e12·25-s − 5.64e12·27-s − 3.81e12·29-s − 4.90e12·31-s + 7.07e12·33-s + 1.21e13·35-s + 1.36e13·37-s − 5.20e13·39-s + 1.03e13·41-s + 1.25e14·43-s + 2.27e14·45-s − 2.52e13·47-s + 3.32e14·49-s + 7.81e14·51-s − 2.39e14·53-s − 1.42e14·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.604·5-s + 1.51·7-s + 10/3·9-s − 0.379·11-s + 0.674·13-s − 1.39·15-s − 1.03·17-s − 1.23·19-s − 3.49·21-s + 0.102·23-s − 1.97·25-s − 3.84·27-s − 1.41·29-s − 1.03·31-s + 0.875·33-s + 0.914·35-s + 0.639·37-s − 1.55·39-s + 0.201·41-s + 1.63·43-s + 2.01·45-s − 0.154·47-s + 10/7·49-s + 2.38·51-s − 0.528·53-s − 0.229·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+17/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(49787136\) = \(2^{8} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.61084\times 10^{8}\) |
| Root analytic conductor: | \(12.4059\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 49787136,\ (\ :17/2, 17/2, 17/2, 17/2),\ 1)\) |
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 | 2 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + p^{8} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{8} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 528276 T + 356648750548 p T^{2} - 8407058218065036 p^{3} T^{3} + \)\(59\!\cdots\!18\)\( p^{5} T^{4} - 8407058218065036 p^{20} T^{5} + 356648750548 p^{35} T^{6} - 528276 p^{51} T^{7} + p^{68} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 269632020 T + 85412116628082964 p T^{2} + \)\(67\!\cdots\!60\)\( p^{2} T^{3} + \)\(30\!\cdots\!10\)\( p^{3} T^{4} + \)\(67\!\cdots\!60\)\( p^{19} T^{5} + 85412116628082964 p^{35} T^{6} + 269632020 p^{51} T^{7} + p^{68} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 1984752056 T + 24441594740650414924 T^{2} - \)\(37\!\cdots\!76\)\( p T^{3} + \)\(16\!\cdots\!02\)\( p^{2} T^{4} - \)\(37\!\cdots\!76\)\( p^{18} T^{5} + 24441594740650414924 p^{34} T^{6} - 1984752056 p^{51} T^{7} + p^{68} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 1750884708 p T + \)\(18\!\cdots\!48\)\( T^{2} + \)\(29\!\cdots\!76\)\( p T^{3} + \)\(23\!\cdots\!18\)\( T^{4} + \)\(29\!\cdots\!76\)\( p^{18} T^{5} + \)\(18\!\cdots\!48\)\( p^{34} T^{6} + 1750884708 p^{52} T^{7} + p^{68} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 91677835096 T + \)\(12\!\cdots\!60\)\( T^{2} + \)\(10\!\cdots\!48\)\( T^{3} + \)\(95\!\cdots\!06\)\( T^{4} + \)\(10\!\cdots\!48\)\( p^{17} T^{5} + \)\(12\!\cdots\!60\)\( p^{34} T^{6} + 91677835096 p^{51} T^{7} + p^{68} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 38384818716 T + \)\(32\!\cdots\!24\)\( T^{2} + \)\(41\!\cdots\!92\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} + \)\(41\!\cdots\!92\)\( p^{17} T^{5} + \)\(32\!\cdots\!24\)\( p^{34} T^{6} - 38384818716 p^{51} T^{7} + p^{68} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 3815624693904 T + \)\(21\!\cdots\!28\)\( T^{2} + \)\(64\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!70\)\( T^{4} + \)\(64\!\cdots\!08\)\( p^{17} T^{5} + \)\(21\!\cdots\!28\)\( p^{34} T^{6} + 3815624693904 p^{51} T^{7} + p^{68} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 4907739231784 T + \)\(83\!\cdots\!92\)\( T^{2} + \)\(30\!\cdots\!08\)\( T^{3} + \)\(27\!\cdots\!18\)\( T^{4} + \)\(30\!\cdots\!08\)\( p^{17} T^{5} + \)\(83\!\cdots\!92\)\( p^{34} T^{6} + 4907739231784 p^{51} T^{7} + p^{68} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 13670824759208 T + \)\(53\!\cdots\!36\)\( T^{2} - \)\(97\!\cdots\!24\)\( T^{3} + \)\(36\!\cdots\!26\)\( T^{4} - \)\(97\!\cdots\!24\)\( p^{17} T^{5} + \)\(53\!\cdots\!36\)\( p^{34} T^{6} - 13670824759208 p^{51} T^{7} + p^{68} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 10307194403604 T + \)\(74\!\cdots\!92\)\( T^{2} - \)\(52\!\cdots\!52\)\( p T^{3} + \)\(25\!\cdots\!82\)\( T^{4} - \)\(52\!\cdots\!52\)\( p^{18} T^{5} + \)\(74\!\cdots\!92\)\( p^{34} T^{6} - 10307194403604 p^{51} T^{7} + p^{68} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 125363311182368 T + \)\(16\!\cdots\!28\)\( T^{2} - \)\(14\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(14\!\cdots\!64\)\( p^{17} T^{5} + \)\(16\!\cdots\!28\)\( p^{34} T^{6} - 125363311182368 p^{51} T^{7} + p^{68} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 25278931593000 T + \)\(79\!\cdots\!20\)\( T^{2} + \)\(38\!\cdots\!88\)\( T^{3} + \)\(27\!\cdots\!82\)\( T^{4} + \)\(38\!\cdots\!88\)\( p^{17} T^{5} + \)\(79\!\cdots\!20\)\( p^{34} T^{6} + 25278931593000 p^{51} T^{7} + p^{68} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 239725859759304 T + \)\(23\!\cdots\!48\)\( T^{2} + \)\(68\!\cdots\!00\)\( T^{3} + \)\(97\!\cdots\!50\)\( T^{4} + \)\(68\!\cdots\!00\)\( p^{17} T^{5} + \)\(23\!\cdots\!48\)\( p^{34} T^{6} + 239725859759304 p^{51} T^{7} + p^{68} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 1374756208497000 T + \)\(44\!\cdots\!88\)\( T^{2} - \)\(46\!\cdots\!32\)\( T^{3} + \)\(79\!\cdots\!02\)\( T^{4} - \)\(46\!\cdots\!32\)\( p^{17} T^{5} + \)\(44\!\cdots\!88\)\( p^{34} T^{6} - 1374756208497000 p^{51} T^{7} + p^{68} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 652646281638080 T + \)\(63\!\cdots\!92\)\( T^{2} - \)\(25\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!62\)\( T^{4} - \)\(25\!\cdots\!56\)\( p^{17} T^{5} + \)\(63\!\cdots\!92\)\( p^{34} T^{6} - 652646281638080 p^{51} T^{7} + p^{68} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 6617359123302392 T + \)\(45\!\cdots\!16\)\( T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(75\!\cdots\!58\)\( T^{4} - \)\(18\!\cdots\!84\)\( p^{17} T^{5} + \)\(45\!\cdots\!16\)\( p^{34} T^{6} - 6617359123302392 p^{51} T^{7} + p^{68} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 2405715947056548 T + \)\(11\!\cdots\!16\)\( T^{2} - \)\(20\!\cdots\!24\)\( T^{3} + \)\(50\!\cdots\!50\)\( T^{4} - \)\(20\!\cdots\!24\)\( p^{17} T^{5} + \)\(11\!\cdots\!16\)\( p^{34} T^{6} - 2405715947056548 p^{51} T^{7} + p^{68} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 4250185178265824 T + \)\(17\!\cdots\!76\)\( T^{2} - \)\(51\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(51\!\cdots\!92\)\( p^{17} T^{5} + \)\(17\!\cdots\!76\)\( p^{34} T^{6} - 4250185178265824 p^{51} T^{7} + p^{68} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 20052572228473688 T + \)\(55\!\cdots\!48\)\( T^{2} - \)\(93\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!02\)\( T^{4} - \)\(93\!\cdots\!76\)\( p^{17} T^{5} + \)\(55\!\cdots\!48\)\( p^{34} T^{6} - 20052572228473688 p^{51} T^{7} + p^{68} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 5593593799375584 T + \)\(98\!\cdots\!40\)\( T^{2} + \)\(79\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!98\)\( T^{4} + \)\(79\!\cdots\!72\)\( p^{17} T^{5} + \)\(98\!\cdots\!40\)\( p^{34} T^{6} + 5593593799375584 p^{51} T^{7} + p^{68} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1518004216828212 T + \)\(32\!\cdots\!88\)\( T^{2} - \)\(55\!\cdots\!56\)\( T^{3} + \)\(61\!\cdots\!98\)\( T^{4} - \)\(55\!\cdots\!56\)\( p^{17} T^{5} + \)\(32\!\cdots\!88\)\( p^{34} T^{6} - 1518004216828212 p^{51} T^{7} + p^{68} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 102152247026326000 T + \)\(22\!\cdots\!56\)\( T^{2} + \)\(16\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!36\)\( p^{17} T^{5} + \)\(22\!\cdots\!56\)\( p^{34} T^{6} + 102152247026326000 p^{51} T^{7} + p^{68} T^{8} \) | |
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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
−7.904272242528202877145722167230, −7.52179274172623425716862192417, −7.10124306962238875477277372957, −7.07592008423627555142240811976, −6.78511718055350405151563074526, −6.16960044656392772934635213558, −6.07505836120352655166833716387, −5.93575072503531700759803507007, −5.79977742097835850609814336138, −5.12160775719412804882344508677, −5.08133822786649841251148103145, −5.05997886121932003999477398819, −4.66277225275894033129612251303, −4.03410840125946889826577524241, −3.82849267446710926299311174639, −3.82812238269327931067334835237, −3.77647397931594066171084715055, −2.51159904546944883751023350634, −2.42452799219896503521701091120, −2.26974586533544452212533190741, −2.08110794740445322589190403358, −1.48860096730260844241121726820, −1.28195741754126507559403819065, −1.04952833520393372252886571411, −1.01941599091846977445887931734, 0, 0, 0, 0, 1.01941599091846977445887931734, 1.04952833520393372252886571411, 1.28195741754126507559403819065, 1.48860096730260844241121726820, 2.08110794740445322589190403358, 2.26974586533544452212533190741, 2.42452799219896503521701091120, 2.51159904546944883751023350634, 3.77647397931594066171084715055, 3.82812238269327931067334835237, 3.82849267446710926299311174639, 4.03410840125946889826577524241, 4.66277225275894033129612251303, 5.05997886121932003999477398819, 5.08133822786649841251148103145, 5.12160775719412804882344508677, 5.79977742097835850609814336138, 5.93575072503531700759803507007, 6.07505836120352655166833716387, 6.16960044656392772934635213558, 6.78511718055350405151563074526, 7.07592008423627555142240811976, 7.10124306962238875477277372957, 7.52179274172623425716862192417, 7.904272242528202877145722167230