Dirichlet series
| L(s) = 1 | − 512·2-s − 1.20e3·3-s + 1.63e5·4-s − 3.12e5·5-s + 6.17e5·6-s + 3.29e6·7-s − 4.19e7·8-s − 2.11e7·9-s + 1.60e8·10-s − 6.80e7·11-s − 1.97e8·12-s − 5.01e7·13-s − 1.68e9·14-s + 3.77e8·15-s + 9.39e9·16-s − 4.22e8·17-s + 1.08e10·18-s − 1.36e9·19-s − 5.12e10·20-s − 3.97e9·21-s + 3.48e10·22-s + 7.13e9·23-s + 5.06e10·24-s + 6.10e10·25-s + 2.56e10·26-s + 1.96e10·27-s + 5.39e11·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.318·3-s + 5·4-s − 1.78·5-s + 0.901·6-s + 1.51·7-s − 7.07·8-s − 1.47·9-s + 5.05·10-s − 1.05·11-s − 1.59·12-s − 0.221·13-s − 4.27·14-s + 0.569·15-s + 35/4·16-s − 0.249·17-s + 4.16·18-s − 0.350·19-s − 8.94·20-s − 0.481·21-s + 2.97·22-s + 0.436·23-s + 2.25·24-s + 2·25-s + 0.627·26-s + 0.360·27-s + 7.55·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(24010000\) = \(2^{4} \cdot 5^{4} \cdot 7^{4}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.95426\times 10^{7}\) |
| Root analytic conductor: | \(9.99427\) |
| Motivic weight: | \(15\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 24010000,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\) |
Particular Values
| \(L(8)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{17}{2})\) | 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 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + p^{7} T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - p^{7} T )^{4} \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 1207 T + 2509399 p^{2} T^{2} + 3682342844 p^{2} T^{3} + 1881192809392 p^{5} T^{4} + 3682342844 p^{17} T^{5} + 2509399 p^{32} T^{6} + 1207 p^{45} T^{7} + p^{60} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 68006431 T + 12727496774589099 T^{2} + \)\(59\!\cdots\!88\)\( p T^{3} + \)\(59\!\cdots\!76\)\( p^{2} T^{4} + \)\(59\!\cdots\!88\)\( p^{16} T^{5} + 12727496774589099 p^{30} T^{6} + 68006431 p^{45} T^{7} + p^{60} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 50184387 T + 7110397241128289 p T^{2} + \)\(23\!\cdots\!90\)\( p^{2} T^{3} + \)\(29\!\cdots\!98\)\( p^{3} T^{4} + \)\(23\!\cdots\!90\)\( p^{17} T^{5} + 7110397241128289 p^{31} T^{6} + 50184387 p^{45} T^{7} + p^{60} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 422044517 T + 8942847953919328545 T^{2} + \)\(31\!\cdots\!38\)\( T^{3} + \)\(34\!\cdots\!90\)\( T^{4} + \)\(31\!\cdots\!38\)\( p^{15} T^{5} + 8942847953919328545 p^{30} T^{6} + 422044517 p^{45} T^{7} + p^{60} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 1365297526 T + 31447782579194999128 T^{2} + \)\(43\!\cdots\!30\)\( p T^{3} + \)\(52\!\cdots\!02\)\( T^{4} + \)\(43\!\cdots\!30\)\( p^{16} T^{5} + 31447782579194999128 p^{30} T^{6} + 1365297526 p^{45} T^{7} + p^{60} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 7134723410 T + \)\(48\!\cdots\!28\)\( T^{2} + \)\(14\!\cdots\!90\)\( T^{3} + \)\(11\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!90\)\( p^{15} T^{5} + \)\(48\!\cdots\!28\)\( p^{30} T^{6} - 7134723410 p^{45} T^{7} + p^{60} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 6569194827 T + \)\(17\!\cdots\!29\)\( p T^{2} + \)\(42\!\cdots\!18\)\( T^{3} - \)\(14\!\cdots\!90\)\( T^{4} + \)\(42\!\cdots\!18\)\( p^{15} T^{5} + \)\(17\!\cdots\!29\)\( p^{31} T^{6} - 6569194827 p^{45} T^{7} + p^{60} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 90078405564 T + \)\(70\!\cdots\!16\)\( T^{2} + \)\(39\!\cdots\!72\)\( T^{3} + \)\(22\!\cdots\!98\)\( T^{4} + \)\(39\!\cdots\!72\)\( p^{15} T^{5} + \)\(70\!\cdots\!16\)\( p^{30} T^{6} + 90078405564 p^{45} T^{7} + p^{60} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 106365934428 T + \)\(55\!\cdots\!60\)\( T^{2} + \)\(18\!\cdots\!92\)\( T^{3} + \)\(11\!\cdots\!22\)\( T^{4} + \)\(18\!\cdots\!92\)\( p^{15} T^{5} + \)\(55\!\cdots\!60\)\( p^{30} T^{6} - 106365934428 p^{45} T^{7} + p^{60} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 + 1911325307330 T + \)\(50\!\cdots\!28\)\( T^{2} + \)\(63\!\cdots\!98\)\( T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(63\!\cdots\!98\)\( p^{15} T^{5} + \)\(50\!\cdots\!28\)\( p^{30} T^{6} + 1911325307330 p^{45} T^{7} + p^{60} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 - 714367563970 T + \)\(51\!\cdots\!52\)\( T^{2} - \)\(61\!\cdots\!62\)\( T^{3} + \)\(22\!\cdots\!42\)\( T^{4} - \)\(61\!\cdots\!62\)\( p^{15} T^{5} + \)\(51\!\cdots\!52\)\( p^{30} T^{6} - 714367563970 p^{45} T^{7} + p^{60} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 30485892979 p T + \)\(14\!\cdots\!27\)\( T^{2} - \)\(21\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!08\)\( T^{4} - \)\(21\!\cdots\!24\)\( p^{15} T^{5} + \)\(14\!\cdots\!27\)\( p^{30} T^{6} - 30485892979 p^{46} T^{7} + p^{60} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 15713425221210 T + \)\(22\!\cdots\!84\)\( T^{2} - \)\(18\!\cdots\!46\)\( T^{3} + \)\(17\!\cdots\!90\)\( T^{4} - \)\(18\!\cdots\!46\)\( p^{15} T^{5} + \)\(22\!\cdots\!84\)\( p^{30} T^{6} - 15713425221210 p^{45} T^{7} + p^{60} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 41124304203296 T + \)\(13\!\cdots\!68\)\( T^{2} - \)\(32\!\cdots\!72\)\( T^{3} + \)\(73\!\cdots\!70\)\( T^{4} - \)\(32\!\cdots\!72\)\( p^{15} T^{5} + \)\(13\!\cdots\!68\)\( p^{30} T^{6} - 41124304203296 p^{45} T^{7} + p^{60} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 82856764898850 T + \)\(40\!\cdots\!20\)\( T^{2} - \)\(14\!\cdots\!54\)\( T^{3} + \)\(41\!\cdots\!50\)\( T^{4} - \)\(14\!\cdots\!54\)\( p^{15} T^{5} + \)\(40\!\cdots\!20\)\( p^{30} T^{6} - 82856764898850 p^{45} T^{7} + p^{60} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 155202052381480 T + \)\(14\!\cdots\!68\)\( T^{2} - \)\(99\!\cdots\!44\)\( T^{3} + \)\(56\!\cdots\!62\)\( T^{4} - \)\(99\!\cdots\!44\)\( p^{15} T^{5} + \)\(14\!\cdots\!68\)\( p^{30} T^{6} - 155202052381480 p^{45} T^{7} + p^{60} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 + 110283212832896 T + \)\(20\!\cdots\!24\)\( T^{2} + \)\(17\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!46\)\( T^{4} + \)\(17\!\cdots\!88\)\( p^{15} T^{5} + \)\(20\!\cdots\!24\)\( p^{30} T^{6} + 110283212832896 p^{45} T^{7} + p^{60} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 138575828632152 T + \)\(36\!\cdots\!08\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!18\)\( T^{4} + \)\(36\!\cdots\!00\)\( p^{15} T^{5} + \)\(36\!\cdots\!08\)\( p^{30} T^{6} + 138575828632152 p^{45} T^{7} + p^{60} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 + 147417914153381 T + \)\(86\!\cdots\!71\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(34\!\cdots\!00\)\( T^{4} + \)\(12\!\cdots\!28\)\( p^{15} T^{5} + \)\(86\!\cdots\!71\)\( p^{30} T^{6} + 147417914153381 p^{45} T^{7} + p^{60} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 332302712293572 T + \)\(23\!\cdots\!56\)\( T^{2} + \)\(47\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!94\)\( T^{4} + \)\(47\!\cdots\!92\)\( p^{15} T^{5} + \)\(23\!\cdots\!56\)\( p^{30} T^{6} + 332302712293572 p^{45} T^{7} + p^{60} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 253043470040998 T + \)\(64\!\cdots\!84\)\( T^{2} - \)\(12\!\cdots\!42\)\( T^{3} + \)\(18\!\cdots\!50\)\( p T^{4} - \)\(12\!\cdots\!42\)\( p^{15} T^{5} + \)\(64\!\cdots\!84\)\( p^{30} T^{6} - 253043470040998 p^{45} T^{7} + p^{60} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 - 64508720871519 T + \)\(19\!\cdots\!57\)\( T^{2} - \)\(60\!\cdots\!26\)\( T^{3} + \)\(16\!\cdots\!54\)\( T^{4} - \)\(60\!\cdots\!26\)\( p^{15} T^{5} + \)\(19\!\cdots\!57\)\( p^{30} T^{6} - 64508720871519 p^{45} T^{7} + p^{60} T^{8} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.428547468274014420758267163995, −8.234456989368120796459967640842, −7.903279898870097085081374565389, −7.86807540961416394149969057252, −7.55535222523509990256278182663, −7.04178393535667145969268732364, −6.95776666325697493870629434723, −6.62454214841873702087203755042, −6.49913753900246461926667616207, −5.54162881155138408201817615688, −5.40151896071952471627501357399, −5.32306926828419845414850073057, −5.26420921189494264062444167905, −4.33337106586592074568989250656, −4.07321272132376342759836003693, −3.72246023044007553364142260910, −3.57271090847914683905025650189, −2.66733786197313199818576061331, −2.60560067899225895524010647902, −2.46763564951632874493392738004, −2.36229519260593675966704093866, −1.47666386932391565277546859933, −1.34161171736921518469982533484, −0.993398452125220350440906584808, −0.874668125569940079431256408464, 0, 0, 0, 0, 0.874668125569940079431256408464, 0.993398452125220350440906584808, 1.34161171736921518469982533484, 1.47666386932391565277546859933, 2.36229519260593675966704093866, 2.46763564951632874493392738004, 2.60560067899225895524010647902, 2.66733786197313199818576061331, 3.57271090847914683905025650189, 3.72246023044007553364142260910, 4.07321272132376342759836003693, 4.33337106586592074568989250656, 5.26420921189494264062444167905, 5.32306926828419845414850073057, 5.40151896071952471627501357399, 5.54162881155138408201817615688, 6.49913753900246461926667616207, 6.62454214841873702087203755042, 6.95776666325697493870629434723, 7.04178393535667145969268732364, 7.55535222523509990256278182663, 7.86807540961416394149969057252, 7.903279898870097085081374565389, 8.234456989368120796459967640842, 8.428547468274014420758267163995