Dirichlet series
| L(s) = 1 | − 256·2-s − 182·3-s + 4.09e4·4-s − 1.79e3·5-s + 4.65e4·6-s − 5.24e6·8-s − 3.47e6·9-s + 4.58e5·10-s + 8.72e6·11-s − 7.45e6·12-s + 7.19e5·13-s + 3.26e5·15-s + 5.87e8·16-s + 7.94e6·17-s + 8.88e8·18-s + 2.15e8·19-s − 7.34e7·20-s − 2.23e9·22-s + 6.19e7·23-s + 9.54e8·24-s − 1.77e9·25-s − 1.84e8·26-s + 1.85e8·27-s − 3.16e9·29-s − 8.34e7·30-s − 6.11e9·31-s − 6.01e10·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.144·3-s + 5·4-s − 0.0512·5-s + 0.407·6-s − 7.07·8-s − 2.17·9-s + 0.145·10-s + 1.48·11-s − 0.720·12-s + 0.0413·13-s + 0.00739·15-s + 35/4·16-s + 0.0798·17-s + 6.15·18-s + 1.05·19-s − 0.256·20-s − 4.20·22-s + 0.0872·23-s + 1.01·24-s − 1.45·25-s − 0.116·26-s + 0.0923·27-s − 0.987·29-s − 0.0209·30-s − 1.23·31-s − 9.89·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(92236816\) = \(2^{4} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.21950\times 10^{8}\) |
| Root analytic conductor: | \(10.2511\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 92236816,\ (\ :13/2, 13/2, 13/2, 13/2),\ 1)\) |
Particular Values
| \(L(7)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{15}{2})\) | 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 | $C_1$ | \( ( 1 + p^{6} T )^{4} \) |
| 7 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_4$ | \( 1 + 182 T + 1168376 p T^{2} + 40147240 p^{3} T^{3} + 9050253185 p^{6} T^{4} + 40147240 p^{16} T^{5} + 1168376 p^{27} T^{6} + 182 p^{39} T^{7} + p^{52} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 1792 T + 1779089486 T^{2} + 18793602841944 p T^{3} + 5602671373883427 p^{3} T^{4} + 18793602841944 p^{14} T^{5} + 1779089486 p^{26} T^{6} + 1792 p^{39} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 8726914 T + 9279143217376 p T^{2} - 6728094454506734904 p^{2} T^{3} + \)\(35\!\cdots\!95\)\( p^{3} T^{4} - 6728094454506734904 p^{15} T^{5} + 9279143217376 p^{27} T^{6} - 8726914 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 - 719208 T + 661703400953692 T^{2} - \)\(30\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!34\)\( T^{4} - \)\(30\!\cdots\!92\)\( p^{13} T^{5} + 661703400953692 p^{26} T^{6} - 719208 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 - 7943068 T + 12483245330733530 T^{2} + \)\(73\!\cdots\!48\)\( T^{3} + \)\(12\!\cdots\!11\)\( T^{4} + \)\(73\!\cdots\!48\)\( p^{13} T^{5} + 12483245330733530 p^{26} T^{6} - 7943068 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 - 215706806 T + 130178678895595216 T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(70\!\cdots\!21\)\( T^{4} - \)\(17\!\cdots\!40\)\( p^{13} T^{5} + 130178678895595216 p^{26} T^{6} - 215706806 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 61927978 T + 1194691014052042340 T^{2} + \)\(11\!\cdots\!28\)\( T^{3} + \)\(70\!\cdots\!65\)\( T^{4} + \)\(11\!\cdots\!28\)\( p^{13} T^{5} + 1194691014052042340 p^{26} T^{6} - 61927978 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 + 3162923032 T + 22916063388720489116 T^{2} + \)\(10\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} + \)\(10\!\cdots\!52\)\( p^{13} T^{5} + 22916063388720489116 p^{26} T^{6} + 3162923032 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 6113775570 T + 71563098448139718316 T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!05\)\( T^{4} + \)\(21\!\cdots\!32\)\( p^{13} T^{5} + 71563098448139718316 p^{26} T^{6} + 6113775570 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 - 3945652880 T + \)\(37\!\cdots\!22\)\( T^{2} - \)\(45\!\cdots\!72\)\( T^{3} + \)\(91\!\cdots\!87\)\( T^{4} - \)\(45\!\cdots\!72\)\( p^{13} T^{5} + \)\(37\!\cdots\!22\)\( p^{26} T^{6} - 3945652880 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 43189289976 T + \)\(25\!\cdots\!84\)\( T^{2} - \)\(40\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!30\)\( T^{4} - \)\(40\!\cdots\!84\)\( p^{13} T^{5} + \)\(25\!\cdots\!84\)\( p^{26} T^{6} - 43189289976 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 54537062128 T + \)\(53\!\cdots\!52\)\( T^{2} + \)\(27\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} + \)\(27\!\cdots\!32\)\( p^{13} T^{5} + \)\(53\!\cdots\!52\)\( p^{26} T^{6} + 54537062128 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 3141202722 T + \)\(99\!\cdots\!92\)\( T^{2} + \)\(46\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!21\)\( T^{4} + \)\(46\!\cdots\!00\)\( p^{13} T^{5} + \)\(99\!\cdots\!92\)\( p^{26} T^{6} - 3141202722 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 + 149625680376 T + \)\(54\!\cdots\!90\)\( T^{2} + \)\(56\!\cdots\!56\)\( T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + \)\(56\!\cdots\!56\)\( p^{13} T^{5} + \)\(54\!\cdots\!90\)\( p^{26} T^{6} + 149625680376 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 866297313938 T + \)\(48\!\cdots\!28\)\( T^{2} - \)\(19\!\cdots\!96\)\( T^{3} + \)\(69\!\cdots\!37\)\( T^{4} - \)\(19\!\cdots\!96\)\( p^{13} T^{5} + \)\(48\!\cdots\!28\)\( p^{26} T^{6} - 866297313938 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 - 477908594184 T + \)\(72\!\cdots\!10\)\( T^{2} - \)\(23\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} - \)\(23\!\cdots\!36\)\( p^{13} T^{5} + \)\(72\!\cdots\!10\)\( p^{26} T^{6} - 477908594184 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 + 1895501016278 T + \)\(34\!\cdots\!40\)\( T^{2} + \)\(34\!\cdots\!56\)\( T^{3} + \)\(31\!\cdots\!33\)\( T^{4} + \)\(34\!\cdots\!56\)\( p^{13} T^{5} + \)\(34\!\cdots\!40\)\( p^{26} T^{6} + 1895501016278 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 319416336064 T + \)\(11\!\cdots\!92\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{3} + \)\(28\!\cdots\!78\)\( p T^{4} + \)\(18\!\cdots\!52\)\( p^{13} T^{5} + \)\(11\!\cdots\!92\)\( p^{26} T^{6} - 319416336064 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 - 2966596192756 T + \)\(94\!\cdots\!86\)\( T^{2} - \)\(15\!\cdots\!88\)\( T^{3} + \)\(35\!\cdots\!91\)\( p T^{4} - \)\(15\!\cdots\!88\)\( p^{13} T^{5} + \)\(94\!\cdots\!86\)\( p^{26} T^{6} - 2966596192756 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 6505959677634 T + \)\(31\!\cdots\!16\)\( T^{2} - \)\(97\!\cdots\!88\)\( T^{3} + \)\(25\!\cdots\!81\)\( T^{4} - \)\(97\!\cdots\!88\)\( p^{13} T^{5} + \)\(31\!\cdots\!16\)\( p^{26} T^{6} - 6505959677634 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 + 1689908567984 T + \)\(20\!\cdots\!40\)\( T^{2} + \)\(40\!\cdots\!32\)\( T^{3} + \)\(20\!\cdots\!38\)\( T^{4} + \)\(40\!\cdots\!32\)\( p^{13} T^{5} + \)\(20\!\cdots\!40\)\( p^{26} T^{6} + 1689908567984 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 + 9586601667468 T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(56\!\cdots\!44\)\( T^{3} + \)\(34\!\cdots\!03\)\( T^{4} + \)\(56\!\cdots\!44\)\( p^{13} T^{5} + \)\(10\!\cdots\!38\)\( p^{26} T^{6} + 9586601667468 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 22280367655784 T + \)\(39\!\cdots\!28\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(42\!\cdots\!74\)\( T^{4} + \)\(43\!\cdots\!36\)\( p^{13} T^{5} + \)\(39\!\cdots\!28\)\( p^{26} T^{6} + 22280367655784 p^{39} T^{7} + p^{52} T^{8} \) | |
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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
−8.430506646581031018468641800633, −7.901289880389069667466320395744, −7.81754067146198185084370599787, −7.76672555865046380085322772990, −7.21141860513078884487533537626, −6.90108870403152402445738156002, −6.71852220810861043510996117897, −6.41649066319732381402004782308, −6.09150678743583654083064488366, −5.78112951488150930591153094295, −5.53586296741899445828675160012, −5.26959027561737285169949784142, −5.09545052544238734192867281217, −4.18284821418994454134364619691, −3.80148926708249528675945437083, −3.62236274348579870359280684893, −3.51075810607226639995820702148, −2.85884482736389217799119439092, −2.52730714176526999550654507566, −2.37643617659455829937685024457, −2.17711061637322121961553374303, −1.55566581968936858100569481103, −1.25491926689055955766880584681, −1.09736235600723122721039103466, −0.984472752645747690683114196624, 0, 0, 0, 0, 0.984472752645747690683114196624, 1.09736235600723122721039103466, 1.25491926689055955766880584681, 1.55566581968936858100569481103, 2.17711061637322121961553374303, 2.37643617659455829937685024457, 2.52730714176526999550654507566, 2.85884482736389217799119439092, 3.51075810607226639995820702148, 3.62236274348579870359280684893, 3.80148926708249528675945437083, 4.18284821418994454134364619691, 5.09545052544238734192867281217, 5.26959027561737285169949784142, 5.53586296741899445828675160012, 5.78112951488150930591153094295, 6.09150678743583654083064488366, 6.41649066319732381402004782308, 6.71852220810861043510996117897, 6.90108870403152402445738156002, 7.21141860513078884487533537626, 7.76672555865046380085322772990, 7.81754067146198185084370599787, 7.901289880389069667466320395744, 8.430506646581031018468641800633