Dirichlet series
| L(s) = 1 | − 860·3-s − 8.00e4·7-s − 1.69e6·9-s + 4.88e6·11-s − 3.19e7·13-s − 9.24e7·17-s − 1.76e8·19-s + 6.88e7·21-s + 2.84e8·23-s + 2.68e9·27-s + 4.41e9·29-s − 2.53e9·31-s − 4.20e9·33-s − 4.98e9·37-s + 2.74e10·39-s + 2.22e10·41-s − 5.77e10·43-s + 8.91e10·47-s − 9.73e10·49-s + 7.94e10·51-s + 8.18e10·53-s + 1.52e11·57-s + 9.86e10·59-s − 2.95e11·61-s + 1.36e11·63-s + 1.27e11·67-s − 2.44e11·69-s + ⋯ |
| L(s) = 1 | − 0.681·3-s − 0.257·7-s − 1.06·9-s + 0.831·11-s − 1.83·13-s − 0.928·17-s − 0.862·19-s + 0.175·21-s + 0.400·23-s + 1.33·27-s + 1.37·29-s − 0.512·31-s − 0.566·33-s − 0.319·37-s + 1.24·39-s + 0.732·41-s − 1.39·43-s + 1.20·47-s − 1.00·49-s + 0.632·51-s + 0.507·53-s + 0.587·57-s + 0.304·59-s − 0.734·61-s + 0.274·63-s + 0.171·67-s − 0.272·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(8\) |
| Conductor: | \(2^{8} \cdot 5^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.32214\times 10^{8}\) |
| Root analytic conductor: | \(10.3552\) |
| Motivic weight: | \(13\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(4\) |
| Selberg data: | \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\) |
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 | \( 1 \) | |
| 5 | \( 1 \) | ||
| good | 3 | $C_2 \wr S_4$ | \( 1 + 860 T + 812818 p T^{2} + 32297600 p^{3} T^{3} + 6114739903 p^{6} T^{4} + 32297600 p^{16} T^{5} + 812818 p^{27} T^{6} + 860 p^{39} T^{7} + p^{52} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 11440 p T + 103746584500 T^{2} + 2468476891733680 p T^{3} + 32107583325455180902 p^{2} T^{4} + 2468476891733680 p^{14} T^{5} + 103746584500 p^{26} T^{6} + 11440 p^{40} T^{7} + p^{52} T^{8} \) | |
| 11 | $C_2 \wr S_4$ | \( 1 - 4884540 T + 11247911129434 p T^{2} - 3786937919384440320 p^{2} T^{3} + \)\(46\!\cdots\!61\)\( p^{3} T^{4} - 3786937919384440320 p^{15} T^{5} + 11247911129434 p^{27} T^{6} - 4884540 p^{39} T^{7} + p^{52} T^{8} \) | |
| 13 | $C_2 \wr S_4$ | \( 1 + 31900360 T + 64433588103748 p T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!62\)\( T^{4} + \)\(13\!\cdots\!00\)\( p^{13} T^{5} + 64433588103748 p^{27} T^{6} + 31900360 p^{39} T^{7} + p^{52} T^{8} \) | |
| 17 | $C_2 \wr S_4$ | \( 1 + 92400180 T + 25162499125791266 T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!27\)\( T^{4} + \)\(21\!\cdots\!00\)\( p^{13} T^{5} + 25162499125791266 p^{26} T^{6} + 92400180 p^{39} T^{7} + p^{52} T^{8} \) | |
| 19 | $C_2 \wr S_4$ | \( 1 + 176935516 T + 126580259575923382 T^{2} + \)\(17\!\cdots\!88\)\( T^{3} + \)\(76\!\cdots\!95\)\( T^{4} + \)\(17\!\cdots\!88\)\( p^{13} T^{5} + 126580259575923382 p^{26} T^{6} + 176935516 p^{39} T^{7} + p^{52} T^{8} \) | |
| 23 | $C_2 \wr S_4$ | \( 1 - 284192040 T + 1343281769458163300 T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!78\)\( T^{4} - \)\(27\!\cdots\!20\)\( p^{13} T^{5} + 1343281769458163300 p^{26} T^{6} - 284192040 p^{39} T^{7} + p^{52} T^{8} \) | |
| 29 | $C_2 \wr S_4$ | \( 1 - 4419104664 T + 36695617526395930292 T^{2} - \)\(10\!\cdots\!72\)\( T^{3} + \)\(52\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!72\)\( p^{13} T^{5} + 36695617526395930292 p^{26} T^{6} - 4419104664 p^{39} T^{7} + p^{52} T^{8} \) | |
| 31 | $C_2 \wr S_4$ | \( 1 + 2533018696 T + 76437197031032318020 T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} + \)\(14\!\cdots\!04\)\( p^{13} T^{5} + 76437197031032318020 p^{26} T^{6} + 2533018696 p^{39} T^{7} + p^{52} T^{8} \) | |
| 37 | $C_2 \wr S_4$ | \( 1 + 4989133120 T + \)\(43\!\cdots\!60\)\( T^{2} + \)\(37\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!18\)\( T^{4} + \)\(37\!\cdots\!40\)\( p^{13} T^{5} + \)\(43\!\cdots\!60\)\( p^{26} T^{6} + 4989133120 p^{39} T^{7} + p^{52} T^{8} \) | |
| 41 | $C_2 \wr S_4$ | \( 1 - 22289893524 T + \)\(28\!\cdots\!50\)\( T^{2} - \)\(63\!\cdots\!76\)\( T^{3} + \)\(35\!\cdots\!39\)\( T^{4} - \)\(63\!\cdots\!76\)\( p^{13} T^{5} + \)\(28\!\cdots\!50\)\( p^{26} T^{6} - 22289893524 p^{39} T^{7} + p^{52} T^{8} \) | |
| 43 | $C_2 \wr S_4$ | \( 1 + 57758591080 T + \)\(49\!\cdots\!72\)\( T^{2} + \)\(23\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + \)\(23\!\cdots\!20\)\( p^{13} T^{5} + \)\(49\!\cdots\!72\)\( p^{26} T^{6} + 57758591080 p^{39} T^{7} + p^{52} T^{8} \) | |
| 47 | $C_2 \wr S_4$ | \( 1 - 89103831000 T + \)\(14\!\cdots\!56\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!42\)\( T^{4} - \)\(10\!\cdots\!00\)\( p^{13} T^{5} + \)\(14\!\cdots\!56\)\( p^{26} T^{6} - 89103831000 p^{39} T^{7} + p^{52} T^{8} \) | |
| 53 | $C_2 \wr S_4$ | \( 1 - 81872204280 T + \)\(25\!\cdots\!80\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!58\)\( T^{4} - \)\(23\!\cdots\!40\)\( p^{13} T^{5} + \)\(25\!\cdots\!80\)\( p^{26} T^{6} - 81872204280 p^{39} T^{7} + p^{52} T^{8} \) | |
| 59 | $C_2 \wr S_4$ | \( 1 - 98645174328 T + \)\(16\!\cdots\!60\)\( T^{2} - \)\(75\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!74\)\( T^{4} - \)\(75\!\cdots\!08\)\( p^{13} T^{5} + \)\(16\!\cdots\!60\)\( p^{26} T^{6} - 98645174328 p^{39} T^{7} + p^{52} T^{8} \) | |
| 61 | $C_2 \wr S_4$ | \( 1 + 295401437392 T + \)\(57\!\cdots\!48\)\( T^{2} + \)\(12\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!24\)\( p^{13} T^{5} + \)\(57\!\cdots\!48\)\( p^{26} T^{6} + 295401437392 p^{39} T^{7} + p^{52} T^{8} \) | |
| 67 | $C_2 \wr S_4$ | \( 1 - 127264608980 T + \)\(10\!\cdots\!30\)\( T^{2} - \)\(25\!\cdots\!60\)\( T^{3} + \)\(69\!\cdots\!63\)\( T^{4} - \)\(25\!\cdots\!60\)\( p^{13} T^{5} + \)\(10\!\cdots\!30\)\( p^{26} T^{6} - 127264608980 p^{39} T^{7} + p^{52} T^{8} \) | |
| 71 | $C_2 \wr S_4$ | \( 1 - 640088634312 T + \)\(30\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!04\)\( T^{3} + \)\(47\!\cdots\!70\)\( T^{4} - \)\(15\!\cdots\!04\)\( p^{13} T^{5} + \)\(30\!\cdots\!48\)\( p^{26} T^{6} - 640088634312 p^{39} T^{7} + p^{52} T^{8} \) | |
| 73 | $C_2 \wr S_4$ | \( 1 + 1295389992460 T + \)\(59\!\cdots\!74\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!47\)\( T^{4} + \)\(64\!\cdots\!00\)\( p^{13} T^{5} + \)\(59\!\cdots\!74\)\( p^{26} T^{6} + 1295389992460 p^{39} T^{7} + p^{52} T^{8} \) | |
| 79 | $C_2 \wr S_4$ | \( 1 - 703795452032 T + \)\(60\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(67\!\cdots\!74\)\( T^{4} + \)\(10\!\cdots\!08\)\( p^{13} T^{5} + \)\(60\!\cdots\!40\)\( p^{26} T^{6} - 703795452032 p^{39} T^{7} + p^{52} T^{8} \) | |
| 83 | $C_2 \wr S_4$ | \( 1 - 3459926885700 T + \)\(23\!\cdots\!10\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!63\)\( T^{4} - \)\(43\!\cdots\!00\)\( p^{13} T^{5} + \)\(23\!\cdots\!10\)\( p^{26} T^{6} - 3459926885700 p^{39} T^{7} + p^{52} T^{8} \) | |
| 89 | $C_2 \wr S_4$ | \( 1 - 3774251627316 T + \)\(44\!\cdots\!22\)\( T^{2} - \)\(21\!\cdots\!68\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(21\!\cdots\!68\)\( p^{13} T^{5} + \)\(44\!\cdots\!22\)\( p^{26} T^{6} - 3774251627316 p^{39} T^{7} + p^{52} T^{8} \) | |
| 97 | $C_2 \wr S_4$ | \( 1 + 18781332004360 T + \)\(36\!\cdots\!40\)\( T^{2} + \)\(38\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!58\)\( T^{4} + \)\(38\!\cdots\!20\)\( p^{13} T^{5} + \)\(36\!\cdots\!40\)\( p^{26} T^{6} + 18781332004360 p^{39} T^{7} + p^{52} T^{8} \) | |
| 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
−8.429030425821564127988216206084, −7.71183703990322237893369828435, −7.65564322826316496666960547278, −7.55288287665023484652080134934, −6.81108968596099164613003627148, −6.70488247628057738829543196551, −6.57144389218296162253288583505, −6.25386786141801806948017582940, −6.14325079794183589845078753875, −5.35871508747836136601409915326, −5.30420563585082306533918372525, −5.14297360055982782375995137980, −4.88146401819062997798563548939, −4.41586738578845783565997897930, −4.07080441756216551418986495494, −3.76087222947135016125029838134, −3.72130567031386121649202032741, −2.79331420940464816183438090570, −2.73415874970092544985335164849, −2.58838264787473698470994469647, −2.43376762570843781976776474431, −1.78246866449982529570106626110, −1.36358602946358851037190192994, −1.18607046215170300868088374576, −0.890735057768836370086007643673, 0, 0, 0, 0, 0.890735057768836370086007643673, 1.18607046215170300868088374576, 1.36358602946358851037190192994, 1.78246866449982529570106626110, 2.43376762570843781976776474431, 2.58838264787473698470994469647, 2.73415874970092544985335164849, 2.79331420940464816183438090570, 3.72130567031386121649202032741, 3.76087222947135016125029838134, 4.07080441756216551418986495494, 4.41586738578845783565997897930, 4.88146401819062997798563548939, 5.14297360055982782375995137980, 5.30420563585082306533918372525, 5.35871508747836136601409915326, 6.14325079794183589845078753875, 6.25386786141801806948017582940, 6.57144389218296162253288583505, 6.70488247628057738829543196551, 6.81108968596099164613003627148, 7.55288287665023484652080134934, 7.65564322826316496666960547278, 7.71183703990322237893369828435, 8.429030425821564127988216206084