Dirichlet series
| L(s) = 1 | + 256·2-s + 1.73e3·3-s + 4.09e4·4-s + 6.25e4·5-s + 4.42e5·6-s + 4.70e5·7-s + 5.24e6·8-s + 5.97e5·9-s + 1.60e7·10-s + 7.46e6·11-s + 7.08e7·12-s + 3.55e7·13-s + 1.20e8·14-s + 1.08e8·15-s + 5.87e8·16-s − 7.52e7·17-s + 1.53e8·18-s + 3.58e7·19-s + 2.56e9·20-s + 8.14e8·21-s + 1.91e9·22-s + 8.33e7·23-s + 9.07e9·24-s + 2.44e9·25-s + 9.09e9·26-s − 1.17e9·27-s + 1.92e10·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.37·3-s + 5·4-s + 1.78·5-s + 3.87·6-s + 1.51·7-s + 7.07·8-s + 0.375·9-s + 5.05·10-s + 1.27·11-s + 6.85·12-s + 2.04·13-s + 4.27·14-s + 2.45·15-s + 35/4·16-s − 0.755·17-s + 1.06·18-s + 0.174·19-s + 8.94·20-s + 2.07·21-s + 3.59·22-s + 0.117·23-s + 9.68·24-s + 2·25-s + 5.77·26-s − 0.584·27-s + 7.55·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(24010000\) = \(2^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.17447\times 10^{7}\) |
| Root analytic conductor: | \(8.66381\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((8,\ 24010000,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(\approx\) | \(468.9328160\) |
| \(L(\frac12)\) | \(\approx\) | \(468.9328160\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{6} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 1730 T + 798335 p T^{2} - 71546870 p^{3} T^{3} + 4423882252 p^{6} T^{4} - 71546870 p^{16} T^{5} + 798335 p^{27} T^{6} - 1730 p^{39} T^{7} + p^{52} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 7468070 T + 6573867489471 p T^{2} - 3836988733495341030 p^{2} T^{3} + \)\(20\!\cdots\!76\)\( p^{3} T^{4} - 3836988733495341030 p^{15} T^{5} + 6573867489471 p^{27} T^{6} - 7468070 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 2732706 p T + 1431542875584209 T^{2} - \)\(23\!\cdots\!82\)\( p T^{3} + \)\(67\!\cdots\!80\)\( T^{4} - \)\(23\!\cdots\!82\)\( p^{14} T^{5} + 1431542875584209 p^{26} T^{6} - 2732706 p^{40} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 75233882 T + 219036778150857 T^{2} + \)\(17\!\cdots\!50\)\( T^{3} + \)\(44\!\cdots\!76\)\( T^{4} + \)\(17\!\cdots\!50\)\( p^{13} T^{5} + 219036778150857 p^{26} T^{6} + 75233882 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 35864396 T + 134801086965369760 T^{2} - \)\(37\!\cdots\!52\)\( T^{3} + \)\(80\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!52\)\( p^{13} T^{5} + 134801086965369760 p^{26} T^{6} - 35864396 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 83365172 T + 1706303976391801248 T^{2} - \)\(12\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!90\)\( T^{4} - \)\(12\!\cdots\!48\)\( p^{13} T^{5} + 1706303976391801248 p^{26} T^{6} - 83365172 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 29892642 p T + 36140008765726644497 T^{2} - \)\(26\!\cdots\!46\)\( T^{3} + \)\(53\!\cdots\!24\)\( T^{4} - \)\(26\!\cdots\!46\)\( p^{13} T^{5} + 36140008765726644497 p^{26} T^{6} - 29892642 p^{40} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 - 7108018656 T + 80064015630926716892 T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!66\)\( T^{4} - \)\(41\!\cdots\!20\)\( p^{13} T^{5} + 80064015630926716892 p^{26} T^{6} - 7108018656 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 9539871312 T + \)\(62\!\cdots\!92\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!86\)\( T^{4} + \)\(46\!\cdots\!00\)\( p^{13} T^{5} + \)\(62\!\cdots\!92\)\( p^{26} T^{6} + 9539871312 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 11228759716 T + \)\(25\!\cdots\!92\)\( T^{2} - \)\(35\!\cdots\!60\)\( T^{3} + \)\(31\!\cdots\!66\)\( T^{4} - \)\(35\!\cdots\!60\)\( p^{13} T^{5} + \)\(25\!\cdots\!92\)\( p^{26} T^{6} - 11228759716 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 16571051476 T + \)\(66\!\cdots\!60\)\( T^{2} - \)\(79\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!46\)\( T^{4} - \)\(79\!\cdots\!76\)\( p^{13} T^{5} + \)\(66\!\cdots\!60\)\( p^{26} T^{6} - 16571051476 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 161745700346 T + \)\(20\!\cdots\!01\)\( T^{2} - \)\(16\!\cdots\!98\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(16\!\cdots\!98\)\( p^{13} T^{5} + \)\(20\!\cdots\!01\)\( p^{26} T^{6} - 161745700346 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 7386754356 T + \)\(47\!\cdots\!20\)\( T^{2} - \)\(25\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!46\)\( T^{4} - \)\(25\!\cdots\!96\)\( p^{13} T^{5} + \)\(47\!\cdots\!20\)\( p^{26} T^{6} - 7386754356 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 234221265584 T - \)\(60\!\cdots\!00\)\( T^{2} - \)\(64\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!18\)\( T^{4} - \)\(64\!\cdots\!48\)\( p^{13} T^{5} - \)\(60\!\cdots\!00\)\( p^{26} T^{6} - 234221265584 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 121559071956 T + \)\(14\!\cdots\!28\)\( T^{2} - \)\(48\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!14\)\( T^{4} - \)\(48\!\cdots\!32\)\( p^{13} T^{5} + \)\(14\!\cdots\!28\)\( p^{26} T^{6} + 121559071956 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 309590851408 T + \)\(16\!\cdots\!72\)\( T^{2} - \)\(37\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(37\!\cdots\!20\)\( p^{13} T^{5} + \)\(16\!\cdots\!72\)\( p^{26} T^{6} - 309590851408 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 1739753267416 T + \)\(48\!\cdots\!92\)\( p T^{2} - \)\(31\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!06\)\( T^{4} - \)\(31\!\cdots\!20\)\( p^{13} T^{5} + \)\(48\!\cdots\!92\)\( p^{27} T^{6} - 1739753267416 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 626488078632 T + \)\(42\!\cdots\!68\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(85\!\cdots\!10\)\( T^{4} - \)\(42\!\cdots\!48\)\( p^{13} T^{5} + \)\(42\!\cdots\!68\)\( p^{26} T^{6} - 626488078632 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 1500472602898 T + \)\(18\!\cdots\!17\)\( T^{2} - \)\(20\!\cdots\!06\)\( T^{3} + \)\(12\!\cdots\!84\)\( T^{4} - \)\(20\!\cdots\!06\)\( p^{13} T^{5} + \)\(18\!\cdots\!17\)\( p^{26} T^{6} - 1500472602898 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 6263474439000 T + \)\(35\!\cdots\!20\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!38\)\( T^{4} - \)\(12\!\cdots\!00\)\( p^{13} T^{5} + \)\(35\!\cdots\!20\)\( p^{26} T^{6} - 6263474439000 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 1169508703900 T + \)\(61\!\cdots\!76\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!66\)\( T^{4} - \)\(18\!\cdots\!00\)\( p^{13} T^{5} + \)\(61\!\cdots\!76\)\( p^{26} T^{6} - 1169508703900 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 10858410211374 T + \)\(11\!\cdots\!97\)\( T^{2} + \)\(21\!\cdots\!66\)\( T^{3} - \)\(11\!\cdots\!20\)\( T^{4} + \)\(21\!\cdots\!66\)\( p^{13} T^{5} + \)\(11\!\cdots\!97\)\( p^{26} T^{6} - 10858410211374 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−8.394038490534381886574120976454, −7.66677623715970160459622078715, −7.41281083534783875346123796357, −7.21364560719469010043823606366, −6.76366954218999176113472774677, −6.30991253040846485385875246362, −6.06455970892673103219986552042, −6.02501058219138552456168868947, −5.92826083436451529171040054818, −5.08078907362261020137536132584, −5.06007533761228681596435967699, −4.80099626352014893016709468598, −4.43502532847173399090562780638, −3.99035902615360425084038328508, −3.60896091477506513261455569520, −3.57559055440160773679204851475, −3.29875898342389821885205798962, −2.56491888355368821627055183424, −2.35101893046591341535354178266, −2.28656155927075384580274244019, −2.03984759990154855933525686994, −1.48443460708683905709023131230, −1.24487679234115582096667314055, −0.959092135604898334252634244534, −0.75661184236197506458746460088, 0.75661184236197506458746460088, 0.959092135604898334252634244534, 1.24487679234115582096667314055, 1.48443460708683905709023131230, 2.03984759990154855933525686994, 2.28656155927075384580274244019, 2.35101893046591341535354178266, 2.56491888355368821627055183424, 3.29875898342389821885205798962, 3.57559055440160773679204851475, 3.60896091477506513261455569520, 3.99035902615360425084038328508, 4.43502532847173399090562780638, 4.80099626352014893016709468598, 5.06007533761228681596435967699, 5.08078907362261020137536132584, 5.92826083436451529171040054818, 6.02501058219138552456168868947, 6.06455970892673103219986552042, 6.30991253040846485385875246362, 6.76366954218999176113472774677, 7.21364560719469010043823606366, 7.41281083534783875346123796357, 7.66677623715970160459622078715, 8.394038490534381886574120976454