Dirichlet series
| L(s) = 1 | + 2.91e3·3-s + 1.92e4·5-s + 4.70e5·7-s + 5.31e6·9-s + 5.15e5·11-s + 3.14e7·13-s + 5.62e7·15-s + 4.04e7·17-s + 2.57e8·19-s + 1.37e9·21-s − 4.92e8·23-s − 1.68e9·25-s + 7.74e9·27-s + 8.17e8·29-s + 1.31e8·31-s + 1.50e9·33-s + 9.07e9·35-s + 3.84e10·37-s + 9.17e10·39-s + 4.24e10·41-s + 4.95e10·43-s + 1.02e11·45-s − 1.52e10·47-s + 1.38e11·49-s + 1.18e11·51-s + 1.89e11·53-s + 9.94e9·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.551·5-s + 1.51·7-s + 10/3·9-s + 0.0878·11-s + 1.80·13-s + 1.27·15-s + 0.406·17-s + 1.25·19-s + 3.49·21-s − 0.693·23-s − 1.38·25-s + 3.84·27-s + 0.255·29-s + 0.0266·31-s + 0.202·33-s + 0.834·35-s + 2.46·37-s + 4.17·39-s + 1.39·41-s + 1.19·43-s + 1.83·45-s − 0.205·47-s + 10/7·49-s + 0.939·51-s + 1.17·53-s + 0.0484·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(49787136\) = \(2^{8} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.58259\times 10^{7}\) |
| Root analytic conductor: | \(9.49073\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 49787136,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(87.04080312\) |
| \(L(\frac12)\) | \(\approx\) | \(87.04080312\) |
| \(L(\frac{15}{2})\) | 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 | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 19278 T + 2056305476 T^{2} - 5852343185802 p T^{3} + 5382954109471254 p^{4} T^{4} - 5852343185802 p^{14} T^{5} + 2056305476 p^{26} T^{6} - 19278 p^{39} T^{7} + p^{52} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 515970 T + 2898349788640 p T^{2} + 1325159351442244854 p^{2} T^{3} + \)\(97\!\cdots\!50\)\( p^{3} T^{4} + 1325159351442244854 p^{15} T^{5} + 2898349788640 p^{27} T^{6} - 515970 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 31470740 T + 75121991670052 p T^{2} - \)\(23\!\cdots\!68\)\( T^{3} + \)\(43\!\cdots\!50\)\( T^{4} - \)\(23\!\cdots\!68\)\( p^{13} T^{5} + 75121991670052 p^{27} T^{6} - 31470740 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 40485258 T + 2109928348081076 T^{2} - \)\(15\!\cdots\!34\)\( T^{3} + \)\(16\!\cdots\!86\)\( T^{4} - \)\(15\!\cdots\!34\)\( p^{13} T^{5} + 2109928348081076 p^{26} T^{6} - 40485258 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 257760344 T + 88377163749427852 T^{2} - \)\(14\!\cdots\!92\)\( T^{3} + \)\(30\!\cdots\!30\)\( T^{4} - \)\(14\!\cdots\!92\)\( p^{13} T^{5} + 88377163749427852 p^{26} T^{6} - 257760344 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 492248178 T + 858325543431583160 T^{2} + \)\(17\!\cdots\!82\)\( T^{3} + \)\(26\!\cdots\!10\)\( T^{4} + \)\(17\!\cdots\!82\)\( p^{13} T^{5} + 858325543431583160 p^{26} T^{6} + 492248178 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 817801920 T + 24595212923825124572 T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(34\!\cdots\!06\)\( T^{4} - \)\(11\!\cdots\!04\)\( p^{13} T^{5} + 24595212923825124572 p^{26} T^{6} - 817801920 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 131625608 T + 80231794136621638204 T^{2} - \)\(45\!\cdots\!04\)\( T^{3} + \)\(27\!\cdots\!66\)\( T^{4} - \)\(45\!\cdots\!04\)\( p^{13} T^{5} + 80231794136621638204 p^{26} T^{6} - 131625608 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 38485873484 T + \)\(57\!\cdots\!48\)\( T^{2} - \)\(62\!\cdots\!08\)\( T^{3} + \)\(92\!\cdots\!38\)\( T^{4} - \)\(62\!\cdots\!08\)\( p^{13} T^{5} + \)\(57\!\cdots\!48\)\( p^{26} T^{6} - 38485873484 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 42425041302 T + \)\(26\!\cdots\!04\)\( T^{2} - \)\(10\!\cdots\!06\)\( T^{3} + \)\(82\!\cdots\!46\)\( p T^{4} - \)\(10\!\cdots\!06\)\( p^{13} T^{5} + \)\(26\!\cdots\!04\)\( p^{26} T^{6} - 42425041302 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 49587853952 T + \)\(54\!\cdots\!92\)\( T^{2} - \)\(19\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!96\)\( p^{13} T^{5} + \)\(54\!\cdots\!92\)\( p^{26} T^{6} - 49587853952 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 15206667516 T + \)\(68\!\cdots\!88\)\( T^{2} - \)\(52\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(52\!\cdots\!08\)\( p^{13} T^{5} + \)\(68\!\cdots\!88\)\( p^{26} T^{6} + 15206667516 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 189707500548 T + \)\(78\!\cdots\!92\)\( T^{2} - \)\(10\!\cdots\!44\)\( T^{3} + \)\(26\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!44\)\( p^{13} T^{5} + \)\(78\!\cdots\!92\)\( p^{26} T^{6} - 189707500548 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 25327021356 T + \)\(18\!\cdots\!08\)\( T^{2} - \)\(59\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!02\)\( T^{4} - \)\(59\!\cdots\!00\)\( p^{13} T^{5} + \)\(18\!\cdots\!08\)\( p^{26} T^{6} - 25327021356 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 57555532688 T + \)\(47\!\cdots\!52\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(34\!\cdots\!80\)\( p^{13} T^{5} + \)\(47\!\cdots\!52\)\( p^{26} T^{6} - 57555532688 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 459346595468 T + \)\(11\!\cdots\!92\)\( T^{2} - \)\(70\!\cdots\!40\)\( T^{3} + \)\(78\!\cdots\!06\)\( T^{4} - \)\(70\!\cdots\!40\)\( p^{13} T^{5} + \)\(11\!\cdots\!92\)\( p^{26} T^{6} - 459346595468 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 510335887698 T + \)\(33\!\cdots\!24\)\( T^{2} - \)\(98\!\cdots\!46\)\( T^{3} + \)\(49\!\cdots\!70\)\( T^{4} - \)\(98\!\cdots\!46\)\( p^{13} T^{5} + \)\(33\!\cdots\!24\)\( p^{26} T^{6} - 510335887698 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 957870191236 T + \)\(55\!\cdots\!64\)\( T^{2} + \)\(41\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!86\)\( T^{4} + \)\(41\!\cdots\!12\)\( p^{13} T^{5} + \)\(55\!\cdots\!64\)\( p^{26} T^{6} + 957870191236 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 21389706452 T + \)\(10\!\cdots\!44\)\( T^{2} - \)\(26\!\cdots\!84\)\( T^{3} + \)\(65\!\cdots\!38\)\( T^{4} - \)\(26\!\cdots\!84\)\( p^{13} T^{5} + \)\(10\!\cdots\!44\)\( p^{26} T^{6} - 21389706452 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 5770833822000 T + \)\(33\!\cdots\!36\)\( T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(38\!\cdots\!10\)\( T^{4} - \)\(10\!\cdots\!72\)\( p^{13} T^{5} + \)\(33\!\cdots\!36\)\( p^{26} T^{6} - 5770833822000 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 7593089475942 T + \)\(63\!\cdots\!16\)\( T^{2} - \)\(40\!\cdots\!74\)\( T^{3} + \)\(17\!\cdots\!86\)\( T^{4} - \)\(40\!\cdots\!74\)\( p^{13} T^{5} + \)\(63\!\cdots\!16\)\( p^{26} T^{6} - 7593089475942 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 5330443230644 T + \)\(18\!\cdots\!00\)\( T^{2} - \)\(91\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!02\)\( T^{4} - \)\(91\!\cdots\!16\)\( p^{13} T^{5} + \)\(18\!\cdots\!00\)\( p^{26} T^{6} - 5330443230644 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−8.178584030520117759403257016427, −7.54154168272555221229169056239, −7.47095502040805029415557440023, −7.38167440290213974172773252842, −7.03894805119653050996325099162, −6.06272692671865616675978472759, −6.05127523718457546434964396285, −6.04520428680912693623156834403, −5.74492903666824667958260116735, −4.94321819892212177132197768504, −4.73779783404932169027844045162, −4.44252152871777333571058791038, −4.29112037641109009658484624703, −3.66160258970293518831178013218, −3.53665366643029309626821454637, −3.29145242993421231638976919098, −3.10046125134519506044605681648, −2.23908589097141634750494962446, −2.14844426366161165824278533886, −2.07959823165243875281944517976, −1.95095373454958707985557097380, −1.15699597495989436612858235470, −0.918537139798492348336430023558, −0.837937434284677960002924574021, −0.66170973766673816892689143754, 0.66170973766673816892689143754, 0.837937434284677960002924574021, 0.918537139798492348336430023558, 1.15699597495989436612858235470, 1.95095373454958707985557097380, 2.07959823165243875281944517976, 2.14844426366161165824278533886, 2.23908589097141634750494962446, 3.10046125134519506044605681648, 3.29145242993421231638976919098, 3.53665366643029309626821454637, 3.66160258970293518831178013218, 4.29112037641109009658484624703, 4.44252152871777333571058791038, 4.73779783404932169027844045162, 4.94321819892212177132197768504, 5.74492903666824667958260116735, 6.04520428680912693623156834403, 6.05127523718457546434964396285, 6.06272692671865616675978472759, 7.03894805119653050996325099162, 7.38167440290213974172773252842, 7.47095502040805029415557440023, 7.54154168272555221229169056239, 8.178584030520117759403257016427