Dirichlet series
| L(s) = 1 | − 64·2-s − 7·3-s + 2.56e3·4-s + 448·6-s − 9.60e3·7-s − 8.19e4·8-s − 3.94e4·9-s − 3.62e4·11-s − 1.79e4·12-s + 1.82e5·13-s + 6.14e5·14-s + 2.29e6·16-s + 3.67e5·17-s + 2.52e6·18-s + 2.22e5·19-s + 6.72e4·21-s + 2.32e6·22-s − 1.82e5·23-s + 5.73e5·24-s − 1.16e7·26-s − 3.76e3·27-s − 2.45e7·28-s + 2.55e5·29-s − 1.22e7·31-s − 5.87e7·32-s + 2.53e5·33-s − 2.35e7·34-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.0498·3-s + 5·4-s + 0.141·6-s − 1.51·7-s − 7.07·8-s − 2.00·9-s − 0.747·11-s − 0.249·12-s + 1.77·13-s + 4.27·14-s + 35/4·16-s + 1.06·17-s + 5.67·18-s + 0.391·19-s + 0.0754·21-s + 2.11·22-s − 0.135·23-s + 0.352·24-s − 5.01·26-s − 0.00136·27-s − 7.55·28-s + 0.0670·29-s − 2.38·31-s − 9.89·32-s + 0.0372·33-s − 3.01·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{4} \cdot 5^{8} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.05589\times 10^{9}\) |
| Root analytic conductor: | \(13.4261\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 2^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(1.346462959\) |
| \(L(\frac12)\) | \(\approx\) | \(1.346462959\) |
| \(L(\frac{11}{2})\) | 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 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| 5 | \( 1 \) | ||
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 7 T + 13174 p T^{2} + 61859 p^{2} T^{3} + 32642414 p^{3} T^{4} + 61859 p^{11} T^{5} + 13174 p^{19} T^{6} + 7 p^{27} T^{7} + p^{36} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 36284 T + 7410570434 T^{2} + 174634924353136 T^{3} + 23267227113398662835 T^{4} + 174634924353136 p^{9} T^{5} + 7410570434 p^{18} T^{6} + 36284 p^{27} T^{7} + p^{36} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 182630 T + 39757005908 T^{2} - 5243881449547314 T^{3} + \)\(61\!\cdots\!02\)\( T^{4} - 5243881449547314 p^{9} T^{5} + 39757005908 p^{18} T^{6} - 182630 p^{27} T^{7} + p^{36} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 367563 T + 422294390492 T^{2} - 113336207887414225 T^{3} + \)\(73\!\cdots\!86\)\( T^{4} - 113336207887414225 p^{9} T^{5} + 422294390492 p^{18} T^{6} - 367563 p^{27} T^{7} + p^{36} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 222229 T + 845335035598 T^{2} - 149198211751817865 T^{3} + \)\(17\!\cdots\!18\)\( p T^{4} - 149198211751817865 p^{9} T^{5} + 845335035598 p^{18} T^{6} - 222229 p^{27} T^{7} + p^{36} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 182439 T + 2075351392807 T^{2} - 95330932645977648 p T^{3} + \)\(14\!\cdots\!44\)\( T^{4} - 95330932645977648 p^{10} T^{5} + 2075351392807 p^{18} T^{6} + 182439 p^{27} T^{7} + p^{36} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 255537 T + 7622478956629 T^{2} - 5519894995589887338 T^{3} + \)\(42\!\cdots\!10\)\( T^{4} - 5519894995589887338 p^{9} T^{5} + 7622478956629 p^{18} T^{6} - 255537 p^{27} T^{7} + p^{36} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 12276460 T + 110687413117548 T^{2} + \)\(68\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!46\)\( T^{4} + \)\(68\!\cdots\!68\)\( p^{9} T^{5} + 110687413117548 p^{18} T^{6} + 12276460 p^{27} T^{7} + p^{36} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 4274163 T + 357194416065487 T^{2} - \)\(10\!\cdots\!70\)\( T^{3} + \)\(62\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!70\)\( p^{9} T^{5} + 357194416065487 p^{18} T^{6} - 4274163 p^{27} T^{7} + p^{36} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 17136315 T + 689062300408418 T^{2} + \)\(16\!\cdots\!73\)\( T^{3} + \)\(26\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!73\)\( p^{9} T^{5} + 689062300408418 p^{18} T^{6} + 17136315 p^{27} T^{7} + p^{36} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 27962067 T - 52834464128425 T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + \)\(45\!\cdots\!40\)\( T^{4} - \)\(44\!\cdots\!08\)\( p^{9} T^{5} - 52834464128425 p^{18} T^{6} - 27962067 p^{27} T^{7} + p^{36} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 26065620 T + 3399149605789532 T^{2} - \)\(89\!\cdots\!92\)\( T^{3} + \)\(51\!\cdots\!62\)\( T^{4} - \)\(89\!\cdots\!92\)\( p^{9} T^{5} + 3399149605789532 p^{18} T^{6} - 26065620 p^{27} T^{7} + p^{36} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 89230902 T + 6156933744974280 T^{2} - \)\(15\!\cdots\!98\)\( T^{3} + \)\(10\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!98\)\( p^{9} T^{5} + 6156933744974280 p^{18} T^{6} - 89230902 p^{27} T^{7} + p^{36} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 96035996 T + 25063423078938988 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!62\)\( T^{4} - \)\(11\!\cdots\!40\)\( p^{9} T^{5} + 25063423078938988 p^{18} T^{6} - 96035996 p^{27} T^{7} + p^{36} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 724808 p T + 11283718331751684 T^{2} + \)\(56\!\cdots\!24\)\( T^{3} + \)\(28\!\cdots\!26\)\( T^{4} + \)\(56\!\cdots\!24\)\( p^{9} T^{5} + 11283718331751684 p^{18} T^{6} + 724808 p^{28} T^{7} + p^{36} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 59945448 T + 82257334571335702 T^{2} - \)\(27\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!31\)\( T^{4} - \)\(27\!\cdots\!80\)\( p^{9} T^{5} + 82257334571335702 p^{18} T^{6} - 59945448 p^{27} T^{7} + p^{36} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 232110635 T + 144366445782369853 T^{2} + \)\(22\!\cdots\!88\)\( T^{3} + \)\(89\!\cdots\!66\)\( T^{4} + \)\(22\!\cdots\!88\)\( p^{9} T^{5} + 144366445782369853 p^{18} T^{6} + 232110635 p^{27} T^{7} + p^{36} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 28740649 T + 161354920336580322 T^{2} - \)\(66\!\cdots\!89\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(66\!\cdots\!89\)\( p^{9} T^{5} + 161354920336580322 p^{18} T^{6} + 28740649 p^{27} T^{7} + p^{36} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 155306887 T + 267975712279382329 T^{2} - \)\(80\!\cdots\!88\)\( T^{3} + \)\(35\!\cdots\!10\)\( T^{4} - \)\(80\!\cdots\!88\)\( p^{9} T^{5} + 267975712279382329 p^{18} T^{6} - 155306887 p^{27} T^{7} + p^{36} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 383782847 T + 268357345969516312 T^{2} + \)\(14\!\cdots\!11\)\( T^{3} + \)\(31\!\cdots\!38\)\( T^{4} + \)\(14\!\cdots\!11\)\( p^{9} T^{5} + 268357345969516312 p^{18} T^{6} + 383782847 p^{27} T^{7} + p^{36} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 988710835 T + 1183023663472280936 T^{2} + \)\(85\!\cdots\!45\)\( T^{3} + \)\(62\!\cdots\!86\)\( T^{4} + \)\(85\!\cdots\!45\)\( p^{9} T^{5} + 1183023663472280936 p^{18} T^{6} + 988710835 p^{27} T^{7} + p^{36} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 950576942 T + 1463971653909090748 T^{2} - \)\(14\!\cdots\!42\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} - \)\(14\!\cdots\!42\)\( p^{9} T^{5} + 1463971653909090748 p^{18} T^{6} - 950576942 p^{27} T^{7} + p^{36} T^{8} \) | |
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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
−7.08523775601141758068415112705, −6.55187775839775099315905323045, −6.24464267256096326596872327347, −6.19494596398466775279321448551, −5.89626854802872160323775710924, −5.68913302679746296412804472787, −5.39332487005384934422505530802, −5.37122527200377952187809600149, −5.07924650185632730806683492669, −4.14353154039122651977182158806, −3.93191870716843490090578876593, −3.73188239474693563370536795093, −3.62384719828782495561365755331, −2.83582557436980021779394001977, −2.83552303825341977394069745811, −2.82272187151604376919665497605, −2.81081023021153587931915452676, −1.92605049045582248420396453866, −1.79956656666023185846880465447, −1.61512810853127610656007818824, −1.27064551738559274487912267021, −0.67799481179007059257313915283, −0.52884972594896416816253348402, −0.44550344623739654124986306215, −0.40642775992555308832946305593, 0.40642775992555308832946305593, 0.44550344623739654124986306215, 0.52884972594896416816253348402, 0.67799481179007059257313915283, 1.27064551738559274487912267021, 1.61512810853127610656007818824, 1.79956656666023185846880465447, 1.92605049045582248420396453866, 2.81081023021153587931915452676, 2.82272187151604376919665497605, 2.83552303825341977394069745811, 2.83582557436980021779394001977, 3.62384719828782495561365755331, 3.73188239474693563370536795093, 3.93191870716843490090578876593, 4.14353154039122651977182158806, 5.07924650185632730806683492669, 5.37122527200377952187809600149, 5.39332487005384934422505530802, 5.68913302679746296412804472787, 5.89626854802872160323775710924, 6.19494596398466775279321448551, 6.24464267256096326596872327347, 6.55187775839775099315905323045, 7.08523775601141758068415112705