Dirichlet series
| L(s) = 1 | + 2.04e3·2-s − 5.88e4·3-s + 2.62e6·4-s + 7.81e6·5-s − 1.20e8·6-s + 1.61e8·7-s + 2.68e9·8-s − 3.82e8·9-s + 1.60e10·10-s + 3.50e9·11-s − 1.54e11·12-s − 3.88e10·13-s + 3.30e11·14-s − 4.60e11·15-s + 2.40e12·16-s − 6.05e11·17-s − 7.83e11·18-s − 3.19e12·19-s + 2.04e13·20-s − 9.50e12·21-s + 7.17e12·22-s − 2.03e13·23-s − 1.58e14·24-s + 3.81e13·25-s − 7.95e13·26-s + 8.15e13·27-s + 4.23e14·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.72·3-s + 5·4-s + 1.78·5-s − 4.88·6-s + 1.51·7-s + 7.07·8-s − 0.329·9-s + 5.05·10-s + 0.447·11-s − 8.63·12-s − 1.01·13-s + 4.27·14-s − 3.09·15-s + 35/4·16-s − 1.23·17-s − 0.930·18-s − 2.27·19-s + 8.94·20-s − 2.61·21-s + 1.26·22-s − 2.36·23-s − 12.2·24-s + 2·25-s − 2.87·26-s + 2.05·27-s + 7.55·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+19/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(24010000\) = \(2^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.58177\times 10^{8}\) |
| Root analytic conductor: | \(12.6558\) |
| Motivic weight: | \(19\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 24010000,\ (\ :19/2, 19/2, 19/2, 19/2),\ 1)\) |
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 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{9} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 19633 p T + 142650889 p^{3} T^{2} + 76750234084 p^{7} T^{3} + 329883404813516 p^{9} T^{4} + 76750234084 p^{26} T^{5} + 142650889 p^{41} T^{6} + 19633 p^{58} T^{7} + p^{76} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 318296981 p T + \)\(20\!\cdots\!59\)\( T^{2} - \)\(59\!\cdots\!24\)\( p T^{3} + \)\(14\!\cdots\!60\)\( p^{2} T^{4} - \)\(59\!\cdots\!24\)\( p^{20} T^{5} + \)\(20\!\cdots\!59\)\( p^{38} T^{6} - 318296981 p^{58} T^{7} + p^{76} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 38853797265 T + \)\(50\!\cdots\!17\)\( T^{2} + \)\(12\!\cdots\!46\)\( p T^{3} + \)\(62\!\cdots\!58\)\( p^{2} T^{4} + \)\(12\!\cdots\!46\)\( p^{20} T^{5} + \)\(50\!\cdots\!17\)\( p^{38} T^{6} + 38853797265 p^{57} T^{7} + p^{76} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 2094528319 p^{2} T + \)\(13\!\cdots\!97\)\( p^{2} T^{2} + \)\(41\!\cdots\!50\)\( p^{3} T^{3} + \)\(15\!\cdots\!66\)\( p^{4} T^{4} + \)\(41\!\cdots\!50\)\( p^{22} T^{5} + \)\(13\!\cdots\!97\)\( p^{40} T^{6} + 2094528319 p^{59} T^{7} + p^{76} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 168299271766 p T + \)\(50\!\cdots\!48\)\( T^{2} + \)\(38\!\cdots\!10\)\( p T^{3} + \)\(11\!\cdots\!42\)\( T^{4} + \)\(38\!\cdots\!10\)\( p^{20} T^{5} + \)\(50\!\cdots\!48\)\( p^{38} T^{6} + 168299271766 p^{58} T^{7} + p^{76} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 + 20389539507398 T + \)\(24\!\cdots\!68\)\( T^{2} + \)\(19\!\cdots\!78\)\( T^{3} + \)\(16\!\cdots\!94\)\( T^{4} + \)\(19\!\cdots\!78\)\( p^{19} T^{5} + \)\(24\!\cdots\!68\)\( p^{38} T^{6} + 20389539507398 p^{57} T^{7} + p^{76} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 71240277291537 T + \)\(36\!\cdots\!01\)\( p T^{2} + \)\(12\!\cdots\!38\)\( T^{3} + \)\(33\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!38\)\( p^{19} T^{5} + \)\(36\!\cdots\!01\)\( p^{39} T^{6} + 71240277291537 p^{57} T^{7} + p^{76} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 52695693915960 T + \)\(68\!\cdots\!48\)\( T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(19\!\cdots\!46\)\( T^{4} + \)\(16\!\cdots\!68\)\( p^{19} T^{5} + \)\(68\!\cdots\!48\)\( p^{38} T^{6} + 52695693915960 p^{57} T^{7} + p^{76} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 1140405782636184 T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!46\)\( T^{4} - \)\(57\!\cdots\!40\)\( p^{19} T^{5} + \)\(17\!\cdots\!88\)\( p^{38} T^{6} - 1140405782636184 p^{57} T^{7} + p^{76} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 1291238635319590 T + \)\(24\!\cdots\!68\)\( T^{2} - \)\(87\!\cdots\!02\)\( T^{3} - \)\(19\!\cdots\!14\)\( T^{4} - \)\(87\!\cdots\!02\)\( p^{19} T^{5} + \)\(24\!\cdots\!68\)\( p^{38} T^{6} + 1291238635319590 p^{57} T^{7} + p^{76} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 4002088743547706 T + \)\(43\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!06\)\( T^{3} + \)\(71\!\cdots\!90\)\( T^{4} + \)\(12\!\cdots\!06\)\( p^{19} T^{5} + \)\(43\!\cdots\!00\)\( p^{38} T^{6} + 4002088743547706 p^{57} T^{7} + p^{76} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 + 16872206497337015 T + \)\(29\!\cdots\!43\)\( T^{2} + \)\(29\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!12\)\( T^{4} + \)\(29\!\cdots\!56\)\( p^{19} T^{5} + \)\(29\!\cdots\!43\)\( p^{38} T^{6} + 16872206497337015 p^{57} T^{7} + p^{76} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 22734909593308386 T + \)\(24\!\cdots\!20\)\( T^{2} + \)\(39\!\cdots\!86\)\( T^{3} + \)\(21\!\cdots\!10\)\( T^{4} + \)\(39\!\cdots\!86\)\( p^{19} T^{5} + \)\(24\!\cdots\!20\)\( p^{38} T^{6} + 22734909593308386 p^{57} T^{7} + p^{76} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 + 123018638929407296 T + \)\(11\!\cdots\!88\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!62\)\( T^{4} + \)\(42\!\cdots\!40\)\( p^{19} T^{5} + \)\(11\!\cdots\!88\)\( p^{38} T^{6} + 123018638929407296 p^{57} T^{7} + p^{76} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 65452640472274062 T + \)\(21\!\cdots\!84\)\( T^{2} - \)\(11\!\cdots\!26\)\( T^{3} + \)\(25\!\cdots\!26\)\( T^{4} - \)\(11\!\cdots\!26\)\( p^{19} T^{5} + \)\(21\!\cdots\!84\)\( p^{38} T^{6} - 65452640472274062 p^{57} T^{7} + p^{76} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 196856787425162164 T + \)\(10\!\cdots\!48\)\( T^{2} - \)\(19\!\cdots\!60\)\( T^{3} + \)\(80\!\cdots\!06\)\( T^{4} - \)\(19\!\cdots\!60\)\( p^{19} T^{5} + \)\(10\!\cdots\!48\)\( p^{38} T^{6} - 196856787425162164 p^{57} T^{7} + p^{76} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 255158367501608840 T + \)\(54\!\cdots\!28\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(10\!\cdots\!12\)\( p^{19} T^{5} + \)\(54\!\cdots\!28\)\( p^{38} T^{6} - 255158367501608840 p^{57} T^{7} + p^{76} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 415730164970581332 T + \)\(43\!\cdots\!28\)\( T^{2} - \)\(32\!\cdots\!24\)\( p T^{3} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(32\!\cdots\!24\)\( p^{20} T^{5} + \)\(43\!\cdots\!28\)\( p^{38} T^{6} - 415730164970581332 p^{57} T^{7} + p^{76} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1170832978362395113 T + \)\(32\!\cdots\!79\)\( T^{2} - \)\(25\!\cdots\!12\)\( T^{3} + \)\(46\!\cdots\!60\)\( T^{4} - \)\(25\!\cdots\!12\)\( p^{19} T^{5} + \)\(32\!\cdots\!79\)\( p^{38} T^{6} - 1170832978362395113 p^{57} T^{7} + p^{76} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 1551039971007315504 T + \)\(69\!\cdots\!88\)\( T^{2} + \)\(11\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!48\)\( p^{19} T^{5} + \)\(69\!\cdots\!88\)\( p^{38} T^{6} + 1551039971007315504 p^{57} T^{7} + p^{76} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 4677744830124879490 T + \)\(32\!\cdots\!36\)\( T^{2} + \)\(11\!\cdots\!30\)\( T^{3} + \)\(53\!\cdots\!86\)\( T^{4} + \)\(11\!\cdots\!30\)\( p^{19} T^{5} + \)\(32\!\cdots\!36\)\( p^{38} T^{6} + 4677744830124879490 p^{57} T^{7} + p^{76} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 16527378829263192819 T + \)\(21\!\cdots\!77\)\( T^{2} + \)\(15\!\cdots\!06\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + \)\(15\!\cdots\!06\)\( p^{19} T^{5} + \)\(21\!\cdots\!77\)\( p^{38} T^{6} + 16527378829263192819 p^{57} T^{7} + p^{76} T^{8} \) | |
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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
−8.000799947294311788643530945550, −7.28062981562964811421478080782, −6.80699610777209669708392623081, −6.77624416944887321712112046854, −6.51946488379188407933483829278, −6.07176416628659971659708193462, −6.05292768272435834214111503155, −5.84332027615052511481657817893, −5.72646794578389567959382154122, −5.08640216030617108453209567942, −4.99281600506425936616419288442, −4.92206356009913995257155099234, −4.79306658296517527846329025272, −4.25570249255189163257428252888, −3.87567759814982125247480579065, −3.78609489740859563626221271001, −3.52190443028485418403736274599, −2.59088150077132089795387008542, −2.49107372917693678936517595895, −2.47068092242674103752863780263, −2.34637833240051575661011733346, −1.71188720687927260977541489802, −1.61861001818856792152243686217, −1.34299290815917625526557311506, −1.17382832554655693511121272464, 0, 0, 0, 0, 1.17382832554655693511121272464, 1.34299290815917625526557311506, 1.61861001818856792152243686217, 1.71188720687927260977541489802, 2.34637833240051575661011733346, 2.47068092242674103752863780263, 2.49107372917693678936517595895, 2.59088150077132089795387008542, 3.52190443028485418403736274599, 3.78609489740859563626221271001, 3.87567759814982125247480579065, 4.25570249255189163257428252888, 4.79306658296517527846329025272, 4.92206356009913995257155099234, 4.99281600506425936616419288442, 5.08640216030617108453209567942, 5.72646794578389567959382154122, 5.84332027615052511481657817893, 6.05292768272435834214111503155, 6.07176416628659971659708193462, 6.51946488379188407933483829278, 6.77624416944887321712112046854, 6.80699610777209669708392623081, 7.28062981562964811421478080782, 8.000799947294311788643530945550