Dirichlet series
| L(s) = 1 | + 64·2-s + 2.56e3·4-s − 320·5-s − 9.60e3·7-s + 8.19e4·8-s − 2.04e4·10-s + 4.43e4·11-s + 7.02e4·13-s − 6.14e5·14-s + 2.29e6·16-s − 5.89e5·17-s − 2.40e5·19-s − 8.19e5·20-s + 2.84e6·22-s + 8.99e5·23-s − 3.95e6·25-s + 4.49e6·26-s − 2.45e7·28-s − 1.36e5·29-s − 5.72e6·31-s + 5.87e7·32-s − 3.77e7·34-s + 3.07e6·35-s − 1.44e7·37-s − 1.53e7·38-s − 2.62e7·40-s + 1.78e7·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.228·5-s − 1.51·7-s + 7.07·8-s − 0.647·10-s + 0.914·11-s + 0.682·13-s − 4.27·14-s + 35/4·16-s − 1.71·17-s − 0.422·19-s − 1.14·20-s + 2.58·22-s + 0.670·23-s − 2.02·25-s + 1.93·26-s − 7.55·28-s − 0.0357·29-s − 1.11·31-s + 9.89·32-s − 4.84·34-s + 0.346·35-s − 1.26·37-s − 1.19·38-s − 1.61·40-s + 0.985·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 3^{12} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.43653\times 10^{9}\) |
| Root analytic conductor: | \(13.9529\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
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 | 2 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| 3 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 64 p T + 4057094 T^{2} + 314786752 p T^{3} + 361650938803 p^{2} T^{4} + 314786752 p^{10} T^{5} + 4057094 p^{18} T^{6} + 64 p^{28} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 44384 T + 8890532894 T^{2} - 293210792391104 T^{3} + 30967739228233791571 T^{4} - 293210792391104 p^{9} T^{5} + 8890532894 p^{18} T^{6} - 44384 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 70296 T + 30403060204 T^{2} - 1413023425292808 T^{3} + 31829252251360759566 p T^{4} - 1413023425292808 p^{9} T^{5} + 30403060204 p^{18} T^{6} - 70296 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 589888 T + 370717081796 T^{2} + 164530489262562496 T^{3} + \)\(66\!\cdots\!38\)\( T^{4} + 164530489262562496 p^{9} T^{5} + 370717081796 p^{18} T^{6} + 589888 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 240100 T + 40331683954 T^{2} - 82510360624689176 T^{3} - \)\(19\!\cdots\!45\)\( T^{4} - 82510360624689176 p^{9} T^{5} + 40331683954 p^{18} T^{6} + 240100 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 899264 T + 4207915482590 T^{2} - 4223151874382778944 T^{3} + \)\(89\!\cdots\!51\)\( T^{4} - 4223151874382778944 p^{9} T^{5} + 4207915482590 p^{18} T^{6} - 899264 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 136352 T + 35176328102684 T^{2} + 31433311519871410016 T^{3} + \)\(63\!\cdots\!02\)\( T^{4} + 31433311519871410016 p^{9} T^{5} + 35176328102684 p^{18} T^{6} + 136352 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 5726484 T + 92634001941874 T^{2} + \)\(36\!\cdots\!76\)\( T^{3} + \)\(34\!\cdots\!15\)\( T^{4} + \)\(36\!\cdots\!76\)\( p^{9} T^{5} + 92634001941874 p^{18} T^{6} + 5726484 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 14425132 T + 347583697782346 T^{2} + \)\(41\!\cdots\!04\)\( T^{3} + \)\(67\!\cdots\!03\)\( T^{4} + \)\(41\!\cdots\!04\)\( p^{9} T^{5} + 347583697782346 p^{18} T^{6} + 14425132 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 17829408 T + 908032461114902 T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(38\!\cdots\!35\)\( T^{4} - \)\(10\!\cdots\!04\)\( p^{9} T^{5} + 908032461114902 p^{18} T^{6} - 17829408 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 28197184 T + 1254502353330316 T^{2} + \)\(35\!\cdots\!96\)\( T^{3} + \)\(74\!\cdots\!78\)\( T^{4} + \)\(35\!\cdots\!96\)\( p^{9} T^{5} + 1254502353330316 p^{18} T^{6} + 28197184 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 1188000 T + 4175107528639964 T^{2} + \)\(20\!\cdots\!84\)\( T^{3} + \)\(68\!\cdots\!30\)\( T^{4} + \)\(20\!\cdots\!84\)\( p^{9} T^{5} + 4175107528639964 p^{18} T^{6} + 1188000 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 38282880 T + 12233793718627412 T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!78\)\( T^{4} + \)\(35\!\cdots\!40\)\( p^{9} T^{5} + 12233793718627412 p^{18} T^{6} + 38282880 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 20787488 T + 27273483779285900 T^{2} + \)\(42\!\cdots\!48\)\( T^{3} + \)\(32\!\cdots\!78\)\( T^{4} + \)\(42\!\cdots\!48\)\( p^{9} T^{5} + 27273483779285900 p^{18} T^{6} + 20787488 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 68357856 T + 16183810930049140 T^{2} - \)\(18\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(18\!\cdots\!96\)\( p^{9} T^{5} + 16183810930049140 p^{18} T^{6} + 68357856 p^{27} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 8117272 T + 27507675872942020 T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!78\)\( T^{4} + \)\(11\!\cdots\!96\)\( p^{9} T^{5} + 27507675872942020 p^{18} T^{6} - 8117272 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 510206080 T + 163125984686865950 T^{2} + \)\(38\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!03\)\( T^{4} + \)\(38\!\cdots\!60\)\( p^{9} T^{5} + 163125984686865950 p^{18} T^{6} + 510206080 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 66623072 T + 59436987599947780 T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!94\)\( T^{4} + \)\(21\!\cdots\!80\)\( p^{9} T^{5} + 59436987599947780 p^{18} T^{6} + 66623072 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 449794224 T + 315976730340925948 T^{2} + \)\(95\!\cdots\!04\)\( T^{3} + \)\(42\!\cdots\!14\)\( T^{4} + \)\(95\!\cdots\!04\)\( p^{9} T^{5} + 315976730340925948 p^{18} T^{6} + 449794224 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 121876384 T + 70578846303035060 T^{2} - \)\(12\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(12\!\cdots\!68\)\( p^{9} T^{5} + 70578846303035060 p^{18} T^{6} + 121876384 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 854372832 T + 633593710691057462 T^{2} + \)\(41\!\cdots\!84\)\( T^{3} + \)\(25\!\cdots\!79\)\( T^{4} + \)\(41\!\cdots\!84\)\( p^{9} T^{5} + 633593710691057462 p^{18} T^{6} + 854372832 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 63831920 T + 760826797991297380 T^{2} + \)\(66\!\cdots\!68\)\( T^{3} + \)\(47\!\cdots\!22\)\( T^{4} + \)\(66\!\cdots\!68\)\( p^{9} T^{5} + 760826797991297380 p^{18} T^{6} + 63831920 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.10013501417347886982221060576, −6.60624270045635644257885945448, −6.48610066240013504286919638545, −6.38502936933590254954196501669, −6.36044636712525545889222372064, −5.71301117858309466417935804040, −5.59708956492038259706847156621, −5.52292316406071624438359111073, −5.41489398409982933613783553260, −4.54651422720771158195001079798, −4.52503721966712087252300761051, −4.52469551427347084381356853192, −4.25743527472046996019773092007, −3.68171984535944737413527225735, −3.57075285240478010617675003503, −3.51567821617903202368234071973, −3.49803492575839586454652887768, −2.71343867872885534590779934969, −2.61618092293537780663731847389, −2.49284098380490547890601943857, −2.21806190656733993656946767831, −1.66535434348719958437032568892, −1.41562935180495892741469331864, −1.29805742453243407847087160277, −1.14867325478687853047117195027, 0, 0, 0, 0, 1.14867325478687853047117195027, 1.29805742453243407847087160277, 1.41562935180495892741469331864, 1.66535434348719958437032568892, 2.21806190656733993656946767831, 2.49284098380490547890601943857, 2.61618092293537780663731847389, 2.71343867872885534590779934969, 3.49803492575839586454652887768, 3.51567821617903202368234071973, 3.57075285240478010617675003503, 3.68171984535944737413527225735, 4.25743527472046996019773092007, 4.52469551427347084381356853192, 4.52503721966712087252300761051, 4.54651422720771158195001079798, 5.41489398409982933613783553260, 5.52292316406071624438359111073, 5.59708956492038259706847156621, 5.71301117858309466417935804040, 6.36044636712525545889222372064, 6.38502936933590254954196501669, 6.48610066240013504286919638545, 6.60624270045635644257885945448, 7.10013501417347886982221060576