Dirichlet series
| L(s) = 1 | − 256·2-s + 2.91e3·3-s + 4.09e4·4-s − 1.50e4·5-s − 7.46e5·6-s − 1.09e5·7-s − 5.24e6·8-s + 5.31e6·9-s + 3.85e6·10-s − 2.82e6·11-s + 1.19e8·12-s − 2.45e7·13-s + 2.81e7·14-s − 4.39e7·15-s + 5.87e8·16-s + 9.65e7·17-s − 1.36e9·18-s + 2.64e8·19-s − 6.17e8·20-s − 3.20e8·21-s + 7.23e8·22-s + 4.58e8·23-s − 1.52e10·24-s − 2.21e9·25-s + 6.29e9·26-s + 7.74e9·27-s − 4.50e9·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 0.431·5-s − 6.53·6-s − 0.353·7-s − 7.07·8-s + 10/3·9-s + 1.21·10-s − 0.481·11-s + 11.5·12-s − 1.41·13-s + 0.998·14-s − 0.995·15-s + 35/4·16-s + 0.970·17-s − 9.42·18-s + 1.28·19-s − 2.15·20-s − 0.815·21-s + 1.36·22-s + 0.645·23-s − 16.3·24-s − 1.81·25-s + 3.99·26-s + 3.84·27-s − 1.76·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{4} \cdot 17^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.43113\times 10^{8}\) |
| Root analytic conductor: | \(10.4582\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{4} \cdot 17^{4} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| 17 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 15066 T + 487600669 p T^{2} + 1922263256394 p^{2} T^{3} + 5287422954302604 p^{4} T^{4} + 1922263256394 p^{15} T^{5} + 487600669 p^{27} T^{6} + 15066 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 109912 T + 217416907324 T^{2} + 4146758766490120 p T^{3} + \)\(55\!\cdots\!74\)\( p^{2} T^{4} + 4146758766490120 p^{14} T^{5} + 217416907324 p^{26} T^{6} + 109912 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 + 2826942 T + 108186751854965 T^{2} + 23589372127093531938 p T^{3} + \)\(42\!\cdots\!88\)\( p^{2} T^{4} + 23589372127093531938 p^{14} T^{5} + 108186751854965 p^{26} T^{6} + 2826942 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 24572254 T + 975424576398073 T^{2} + \)\(19\!\cdots\!50\)\( T^{3} + \)\(42\!\cdots\!76\)\( T^{4} + \)\(19\!\cdots\!50\)\( p^{13} T^{5} + 975424576398073 p^{26} T^{6} + 24572254 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 264166442 T + 108991201738319869 T^{2} - \)\(14\!\cdots\!58\)\( T^{3} + \)\(46\!\cdots\!80\)\( T^{4} - \)\(14\!\cdots\!58\)\( p^{13} T^{5} + 108991201738319869 p^{26} T^{6} - 264166442 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 458114946 T + 1498401569512179365 T^{2} - \)\(61\!\cdots\!94\)\( T^{3} + \)\(10\!\cdots\!72\)\( T^{4} - \)\(61\!\cdots\!94\)\( p^{13} T^{5} + 1498401569512179365 p^{26} T^{6} - 458114946 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 1248566328 T + 15583972207087653788 T^{2} - \)\(42\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!22\)\( T^{4} - \)\(42\!\cdots\!16\)\( p^{13} T^{5} + 15583972207087653788 p^{26} T^{6} - 1248566328 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 2499705268 T + 80775444442292879584 T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!26\)\( T^{4} + \)\(19\!\cdots\!24\)\( p^{13} T^{5} + 80775444442292879584 p^{26} T^{6} + 2499705268 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 33687922732 T + \)\(11\!\cdots\!00\)\( T^{2} + \)\(21\!\cdots\!44\)\( T^{3} + \)\(42\!\cdots\!98\)\( T^{4} + \)\(21\!\cdots\!44\)\( p^{13} T^{5} + \)\(11\!\cdots\!00\)\( p^{26} T^{6} + 33687922732 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 51272609910 T + \)\(40\!\cdots\!93\)\( T^{2} + \)\(12\!\cdots\!58\)\( T^{3} + \)\(56\!\cdots\!76\)\( T^{4} + \)\(12\!\cdots\!58\)\( p^{13} T^{5} + \)\(40\!\cdots\!93\)\( p^{26} T^{6} + 51272609910 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 77668086634 T + \)\(53\!\cdots\!97\)\( T^{2} + \)\(26\!\cdots\!06\)\( T^{3} + \)\(13\!\cdots\!44\)\( T^{4} + \)\(26\!\cdots\!06\)\( p^{13} T^{5} + \)\(53\!\cdots\!97\)\( p^{26} T^{6} + 77668086634 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 58945517700 T + \)\(65\!\cdots\!80\)\( T^{2} + \)\(60\!\cdots\!60\)\( p T^{3} + \)\(40\!\cdots\!18\)\( T^{4} + \)\(60\!\cdots\!60\)\( p^{14} T^{5} + \)\(65\!\cdots\!80\)\( p^{26} T^{6} + 58945517700 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 220825893168 T + \)\(11\!\cdots\!64\)\( T^{2} - \)\(16\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!62\)\( T^{4} - \)\(16\!\cdots\!40\)\( p^{13} T^{5} + \)\(11\!\cdots\!64\)\( p^{26} T^{6} - 220825893168 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 595146273300 T + \)\(40\!\cdots\!92\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(58\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{13} T^{5} + \)\(40\!\cdots\!92\)\( p^{26} T^{6} - 595146273300 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 47251741108 T + \)\(31\!\cdots\!20\)\( T^{2} - \)\(44\!\cdots\!24\)\( T^{3} + \)\(49\!\cdots\!26\)\( T^{4} - \)\(44\!\cdots\!24\)\( p^{13} T^{5} + \)\(31\!\cdots\!20\)\( p^{26} T^{6} + 47251741108 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 516592464560 T + \)\(78\!\cdots\!92\)\( T^{2} - \)\(83\!\cdots\!52\)\( T^{3} + \)\(53\!\cdots\!02\)\( T^{4} - \)\(83\!\cdots\!52\)\( p^{13} T^{5} + \)\(78\!\cdots\!92\)\( p^{26} T^{6} - 516592464560 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 1265743711440 T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!18\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{13} T^{5} + \)\(24\!\cdots\!12\)\( p^{26} T^{6} - 1265743711440 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 1313562407024 T + \)\(55\!\cdots\!16\)\( T^{2} - \)\(44\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(44\!\cdots\!40\)\( p^{13} T^{5} + \)\(55\!\cdots\!16\)\( p^{26} T^{6} - 1313562407024 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 819080451268 T + \)\(13\!\cdots\!12\)\( T^{2} + \)\(31\!\cdots\!84\)\( T^{3} + \)\(81\!\cdots\!54\)\( T^{4} + \)\(31\!\cdots\!84\)\( p^{13} T^{5} + \)\(13\!\cdots\!12\)\( p^{26} T^{6} + 819080451268 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 1211513205492 T + \)\(26\!\cdots\!04\)\( T^{2} - \)\(36\!\cdots\!28\)\( T^{3} + \)\(31\!\cdots\!06\)\( T^{4} - \)\(36\!\cdots\!28\)\( p^{13} T^{5} + \)\(26\!\cdots\!04\)\( p^{26} T^{6} - 1211513205492 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 632791446900 T + \)\(56\!\cdots\!48\)\( T^{2} + \)\(94\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!62\)\( T^{4} + \)\(94\!\cdots\!08\)\( p^{13} T^{5} + \)\(56\!\cdots\!48\)\( p^{26} T^{6} + 632791446900 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 16136347540396 T + \)\(22\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!62\)\( T^{4} + \)\(26\!\cdots\!44\)\( p^{13} T^{5} + \)\(22\!\cdots\!20\)\( p^{26} T^{6} + 16136347540396 p^{39} T^{7} + p^{52} T^{8} \) | |
| 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
−8.436460934685087433629885860577, −7.910685876011428128799784868382, −7.72685841883953083005900057799, −7.62033353240001365843908512021, −7.50074628652001925319192024435, −6.98713917160196771728414934006, −6.67956257272723563135595645879, −6.56842069824271917866239519174, −6.51187463242875587869971777910, −5.46836630553707167148585298487, −5.21656204049017695550442306251, −5.20044325903095933385272910153, −4.91321096059676461212522445520, −3.79041320929986256145034068992, −3.77650574096405485063539047351, −3.54628032324563541648294599827, −3.48235864297541805401968447597, −2.66909837740407859488329198847, −2.61589699045424889324055863809, −2.51659857156321410883754089134, −2.24057254907716729390490488165, −1.55019950731959347835926947863, −1.42028752204851318792854141377, −1.22653943684110740996243750753, −1.16124502096170443135858072093, 0, 0, 0, 0, 1.16124502096170443135858072093, 1.22653943684110740996243750753, 1.42028752204851318792854141377, 1.55019950731959347835926947863, 2.24057254907716729390490488165, 2.51659857156321410883754089134, 2.61589699045424889324055863809, 2.66909837740407859488329198847, 3.48235864297541805401968447597, 3.54628032324563541648294599827, 3.77650574096405485063539047351, 3.79041320929986256145034068992, 4.91321096059676461212522445520, 5.20044325903095933385272910153, 5.21656204049017695550442306251, 5.46836630553707167148585298487, 6.51187463242875587869971777910, 6.56842069824271917866239519174, 6.67956257272723563135595645879, 6.98713917160196771728414934006, 7.50074628652001925319192024435, 7.62033353240001365843908512021, 7.72685841883953083005900057799, 7.910685876011428128799784868382, 8.436460934685087433629885860577