Dirichlet series
| L(s) = 1 | − 7.87e4·3-s − 2.50e6·5-s − 1.61e8·7-s + 3.87e9·9-s − 3.79e9·11-s − 3.91e10·13-s + 1.96e11·15-s − 1.90e11·17-s + 2.29e12·19-s + 1.27e13·21-s − 1.02e13·23-s − 2.97e13·25-s − 1.52e14·27-s − 7.76e13·29-s + 1.96e14·31-s + 2.99e14·33-s + 4.03e14·35-s + 8.29e14·37-s + 3.08e15·39-s + 2.21e15·41-s + 3.70e15·43-s − 9.69e15·45-s + 4.91e15·47-s + 1.62e16·49-s + 1.50e16·51-s + 4.58e15·53-s + 9.50e15·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.572·5-s − 1.51·7-s + 10/3·9-s − 0.485·11-s − 1.02·13-s + 1.32·15-s − 0.390·17-s + 1.63·19-s + 3.49·21-s − 1.19·23-s − 1.55·25-s − 3.84·27-s − 0.994·29-s + 1.33·31-s + 1.12·33-s + 0.866·35-s + 1.04·37-s + 2.36·39-s + 1.05·41-s + 1.12·43-s − 1.90·45-s + 0.640·47-s + 10/7·49-s + 0.901·51-s + 0.190·53-s + 0.278·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(49787136\) = \(2^{8} \cdot 3^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.36479\times 10^{9}\) |
| Root analytic conductor: | \(13.8638\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 49787136,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
Particular Values
| \(L(10)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{21}{2})\) | 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 | $C_1$ | \( ( 1 + p^{9} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + p^{9} T )^{4} \) | |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 100086 p^{2} T + 1440091182588 p^{2} T^{2} + 1017643477750126386 p^{3} T^{3} + \)\(94\!\cdots\!54\)\( p^{7} T^{4} + 1017643477750126386 p^{22} T^{5} + 1440091182588 p^{40} T^{6} + 100086 p^{59} T^{7} + p^{76} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3798155970 T + \)\(21\!\cdots\!00\)\( T^{2} + \)\(59\!\cdots\!54\)\( p T^{3} + \)\(15\!\cdots\!50\)\( p^{2} T^{4} + \)\(59\!\cdots\!54\)\( p^{20} T^{5} + \)\(21\!\cdots\!00\)\( p^{38} T^{6} + 3798155970 p^{57} T^{7} + p^{76} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 39152617980 T + \)\(31\!\cdots\!84\)\( T^{2} + \)\(88\!\cdots\!28\)\( p T^{3} + \)\(41\!\cdots\!50\)\( p^{2} T^{4} + \)\(88\!\cdots\!28\)\( p^{20} T^{5} + \)\(31\!\cdots\!84\)\( p^{38} T^{6} + 39152617980 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 11227802562 p T + \)\(23\!\cdots\!96\)\( p^{2} T^{2} + \)\(74\!\cdots\!46\)\( p^{3} T^{3} + \)\(13\!\cdots\!78\)\( p^{5} T^{4} + \)\(74\!\cdots\!46\)\( p^{22} T^{5} + \)\(23\!\cdots\!96\)\( p^{40} T^{6} + 11227802562 p^{58} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 121012647104 p T + \)\(58\!\cdots\!72\)\( T^{2} - \)\(31\!\cdots\!72\)\( p T^{3} + \)\(10\!\cdots\!30\)\( T^{4} - \)\(31\!\cdots\!72\)\( p^{20} T^{5} + \)\(58\!\cdots\!72\)\( p^{38} T^{6} - 121012647104 p^{58} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 10298330967654 T + \)\(16\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!98\)\( p T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(41\!\cdots\!98\)\( p^{20} T^{5} + \)\(16\!\cdots\!00\)\( p^{38} T^{6} + 10298330967654 p^{57} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 77678357293560 T + \)\(13\!\cdots\!72\)\( T^{2} + \)\(79\!\cdots\!84\)\( T^{3} + \)\(91\!\cdots\!06\)\( T^{4} + \)\(79\!\cdots\!84\)\( p^{19} T^{5} + \)\(13\!\cdots\!72\)\( p^{38} T^{6} + 77678357293560 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 196054920663408 T + \)\(70\!\cdots\!24\)\( T^{2} - \)\(77\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!26\)\( T^{4} - \)\(77\!\cdots\!24\)\( p^{19} T^{5} + \)\(70\!\cdots\!24\)\( p^{38} T^{6} - 196054920663408 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 829369260264428 T + \)\(22\!\cdots\!92\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(21\!\cdots\!58\)\( T^{4} - \)\(14\!\cdots\!44\)\( p^{19} T^{5} + \)\(22\!\cdots\!92\)\( p^{38} T^{6} - 829369260264428 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 2210157258858402 T + \)\(51\!\cdots\!64\)\( T^{2} + \)\(12\!\cdots\!54\)\( T^{3} - \)\(26\!\cdots\!34\)\( T^{4} + \)\(12\!\cdots\!54\)\( p^{19} T^{5} + \)\(51\!\cdots\!64\)\( p^{38} T^{6} - 2210157258858402 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 3704496271875056 T + \)\(41\!\cdots\!88\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(65\!\cdots\!18\)\( T^{4} - \)\(11\!\cdots\!32\)\( p^{19} T^{5} + \)\(41\!\cdots\!88\)\( p^{38} T^{6} - 3704496271875056 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 4910647758864108 T + \)\(15\!\cdots\!32\)\( T^{2} - \)\(10\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} - \)\(10\!\cdots\!44\)\( p^{19} T^{5} + \)\(15\!\cdots\!32\)\( p^{38} T^{6} - 4910647758864108 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 4583139659223684 T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(96\!\cdots\!88\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(96\!\cdots\!88\)\( p^{19} T^{5} + \)\(14\!\cdots\!48\)\( p^{38} T^{6} - 4583139659223684 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 10541900786867604 T + \)\(18\!\cdots\!72\)\( p T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(57\!\cdots\!82\)\( T^{4} + \)\(29\!\cdots\!80\)\( p^{19} T^{5} + \)\(18\!\cdots\!72\)\( p^{39} T^{6} - 10541900786867604 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 67528372286847048 T + \)\(26\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!62\)\( T^{4} - \)\(16\!\cdots\!00\)\( p^{19} T^{5} + \)\(26\!\cdots\!32\)\( p^{38} T^{6} - 67528372286847048 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 232506796904165236 T + \)\(14\!\cdots\!88\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} + \)\(99\!\cdots\!86\)\( T^{4} - \)\(26\!\cdots\!60\)\( p^{19} T^{5} + \)\(14\!\cdots\!88\)\( p^{38} T^{6} - 232506796904165236 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 264200775007920198 T + \)\(24\!\cdots\!44\)\( T^{2} + \)\(56\!\cdots\!54\)\( T^{3} + \)\(64\!\cdots\!30\)\( T^{4} + \)\(56\!\cdots\!54\)\( p^{19} T^{5} + \)\(24\!\cdots\!44\)\( p^{38} T^{6} - 264200775007920198 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 668786810310501652 T + \)\(74\!\cdots\!76\)\( T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!86\)\( T^{4} - \)\(34\!\cdots\!16\)\( p^{19} T^{5} + \)\(74\!\cdots\!76\)\( p^{38} T^{6} - 668786810310501652 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1869350307247472268 T + \)\(37\!\cdots\!84\)\( T^{2} - \)\(51\!\cdots\!56\)\( T^{3} + \)\(57\!\cdots\!18\)\( T^{4} - \)\(51\!\cdots\!56\)\( p^{19} T^{5} + \)\(37\!\cdots\!84\)\( p^{38} T^{6} - 1869350307247472268 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 4190113921662329040 T + \)\(12\!\cdots\!44\)\( T^{2} - \)\(23\!\cdots\!04\)\( T^{3} + \)\(45\!\cdots\!30\)\( T^{4} - \)\(23\!\cdots\!04\)\( p^{19} T^{5} + \)\(12\!\cdots\!44\)\( p^{38} T^{6} - 4190113921662329040 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 4059161776602418782 T + \)\(39\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!94\)\( T^{3} + \)\(63\!\cdots\!06\)\( T^{4} + \)\(11\!\cdots\!94\)\( p^{19} T^{5} + \)\(39\!\cdots\!76\)\( p^{38} T^{6} + 4059161776602418782 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 1033481664034434692 T + \)\(14\!\cdots\!60\)\( T^{2} + \)\(35\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!62\)\( T^{4} + \)\(35\!\cdots\!32\)\( p^{19} T^{5} + \)\(14\!\cdots\!60\)\( p^{38} T^{6} + 1033481664034434692 p^{57} T^{7} + p^{76} T^{8} \) | |
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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
−7.67757836945490432127410328404, −7.25761291038815570278010528403, −6.99569305563629452626497589754, −6.70557072697781635584501855358, −6.59977623795519666695997666079, −6.05493542406534368088034703134, −5.92378762648285142910372643338, −5.82085777159750796628244809079, −5.61156148087272550486843680032, −5.06013955232419705751060160849, −4.97184924970003649556901897127, −4.59626246049821763562303615103, −4.51499744370340301202641335337, −3.81070295798130085790501036394, −3.76552618198158786457556282759, −3.53937457798338369451352090283, −3.49448106920115027314295157939, −2.49100110026320684677426107868, −2.39215171295140132504236649391, −2.33297786011725388223049400332, −2.21630573574308965882978096974, −1.24366525473329302288885970024, −1.05383716657440806534097021497, −1.01080406729600743904082201796, −0.77178985296326797028407348328, 0, 0, 0, 0, 0.77178985296326797028407348328, 1.01080406729600743904082201796, 1.05383716657440806534097021497, 1.24366525473329302288885970024, 2.21630573574308965882978096974, 2.33297786011725388223049400332, 2.39215171295140132504236649391, 2.49100110026320684677426107868, 3.49448106920115027314295157939, 3.53937457798338369451352090283, 3.76552618198158786457556282759, 3.81070295798130085790501036394, 4.51499744370340301202641335337, 4.59626246049821763562303615103, 4.97184924970003649556901897127, 5.06013955232419705751060160849, 5.61156148087272550486843680032, 5.82085777159750796628244809079, 5.92378762648285142910372643338, 6.05493542406534368088034703134, 6.59977623795519666695997666079, 6.70557072697781635584501855358, 6.99569305563629452626497589754, 7.25761291038815570278010528403, 7.67757836945490432127410328404