Dirichlet series
| L(s) = 1 | − 3.35e7·2-s + 2.81e11·3-s + 8.44e14·4-s + 8.31e16·5-s − 9.43e18·6-s − 4.32e20·7-s − 1.88e22·8-s − 2.54e23·9-s − 2.79e24·10-s − 2.50e24·11-s + 2.37e26·12-s − 1.67e27·13-s + 1.45e28·14-s + 2.33e28·15-s + 3.96e29·16-s + 1.24e30·17-s + 8.52e30·18-s + 3.17e31·19-s + 7.02e31·20-s − 1.21e32·21-s + 8.40e31·22-s − 2.87e33·23-s − 5.30e33·24-s − 1.02e34·25-s + 5.61e34·26-s − 9.78e34·27-s − 3.65e35·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.574·3-s + 3/2·4-s + 0.624·5-s − 0.812·6-s − 0.853·7-s − 1.41·8-s − 1.06·9-s − 0.882·10-s − 0.0766·11-s + 0.861·12-s − 0.855·13-s + 1.20·14-s + 0.358·15-s + 5/4·16-s + 0.887·17-s + 1.50·18-s + 1.48·19-s + 0.936·20-s − 0.490·21-s + 0.108·22-s − 1.24·23-s − 0.812·24-s − 0.575·25-s + 1.20·26-s − 0.835·27-s − 1.27·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+49/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(4\) = \(2^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(924.965\) |
| Root analytic conductor: | \(5.51482\) |
| Motivic weight: | \(49\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 4,\ (\ :49/2, 49/2),\ 1)\) |
Particular Values
| \(L(25)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{51}{2})\) | 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 | $C_1$ | \( ( 1 + p^{24} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 10409299096 p^{3} T + 208960133642359634 p^{13} T^{2} - 10409299096 p^{52} T^{3} + p^{98} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 665396249460348 p^{3} T + \)\(43\!\cdots\!26\)\( p^{8} T^{2} - 665396249460348 p^{52} T^{3} + p^{98} T^{4} \) | |
| 7 | $D_{4}$ | \( 1 + \)\(43\!\cdots\!36\)\( T + \)\(96\!\cdots\!66\)\( p^{3} T^{2} + \)\(43\!\cdots\!36\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 11 | $D_{4}$ | \( 1 + \)\(22\!\cdots\!76\)\( p T + \)\(47\!\cdots\!06\)\( p^{5} T^{2} + \)\(22\!\cdots\!76\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 13 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!28\)\( T + \)\(26\!\cdots\!18\)\( p^{2} T^{2} + \)\(16\!\cdots\!28\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 17 | $D_{4}$ | \( 1 - \)\(73\!\cdots\!92\)\( p T + \)\(77\!\cdots\!86\)\( p^{3} T^{2} - \)\(73\!\cdots\!92\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 19 | $D_{4}$ | \( 1 - \)\(31\!\cdots\!20\)\( T + \)\(31\!\cdots\!78\)\( p^{2} T^{2} - \)\(31\!\cdots\!20\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 23 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!36\)\( p T + \)\(71\!\cdots\!66\)\( p^{3} T^{2} + \)\(12\!\cdots\!36\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 29 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!80\)\( T + \)\(34\!\cdots\!22\)\( p T^{2} + \)\(11\!\cdots\!80\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 31 | $D_{4}$ | \( 1 + \)\(71\!\cdots\!96\)\( T + \)\(11\!\cdots\!66\)\( p T^{2} + \)\(71\!\cdots\!96\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 37 | $D_{4}$ | \( 1 + \)\(45\!\cdots\!16\)\( T + \)\(35\!\cdots\!14\)\( p T^{2} + \)\(45\!\cdots\!16\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 41 | $D_{4}$ | \( 1 - \)\(17\!\cdots\!04\)\( p T + \)\(18\!\cdots\!66\)\( p^{2} T^{2} - \)\(17\!\cdots\!04\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 43 | $D_{4}$ | \( 1 - \)\(53\!\cdots\!24\)\( p T + \)\(18\!\cdots\!58\)\( p^{2} T^{2} - \)\(53\!\cdots\!24\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 47 | $D_{4}$ | \( 1 + \)\(25\!\cdots\!68\)\( p T + \)\(49\!\cdots\!82\)\( p^{2} T^{2} + \)\(25\!\cdots\!68\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 53 | $D_{4}$ | \( 1 + \)\(39\!\cdots\!68\)\( T + \)\(99\!\cdots\!22\)\( T^{2} + \)\(39\!\cdots\!68\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 59 | $D_{4}$ | \( 1 - \)\(21\!\cdots\!40\)\( T + \)\(46\!\cdots\!78\)\( T^{2} - \)\(21\!\cdots\!40\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 61 | $D_{4}$ | \( 1 - \)\(57\!\cdots\!04\)\( T + \)\(27\!\cdots\!86\)\( T^{2} - \)\(57\!\cdots\!04\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 67 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!48\)\( p T + \)\(79\!\cdots\!58\)\( T^{2} + \)\(16\!\cdots\!48\)\( p^{50} T^{3} + p^{98} T^{4} \) | |
| 71 | $D_{4}$ | \( 1 + \)\(25\!\cdots\!16\)\( T + \)\(89\!\cdots\!26\)\( T^{2} + \)\(25\!\cdots\!16\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 73 | $D_{4}$ | \( 1 + \)\(97\!\cdots\!88\)\( T + \)\(57\!\cdots\!62\)\( T^{2} + \)\(97\!\cdots\!88\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 79 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!60\)\( T + \)\(12\!\cdots\!38\)\( T^{2} + \)\(16\!\cdots\!60\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 83 | $D_{4}$ | \( 1 + \)\(24\!\cdots\!28\)\( T + \)\(64\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!28\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 89 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(80\!\cdots\!18\)\( T^{2} - \)\(10\!\cdots\!80\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
| 97 | $D_{4}$ | \( 1 - \)\(86\!\cdots\!24\)\( T + \)\(29\!\cdots\!78\)\( T^{2} - \)\(86\!\cdots\!24\)\( p^{49} T^{3} + p^{98} T^{4} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.32505129086880965855091484654, −16.07009390458362862232076917801, −14.56332069409854110068722960153, −14.21704650252904920939280029947, −12.88315956060479463436269766843, −11.95371952249700270094874721146, −11.03093205288358349022703629731, −9.966871464894000697832937261643, −9.413814330920998732272832278168, −8.975210981503701117147819781525, −7.58890240033698295300761005100, −7.47970327921472559890882513517, −5.80957635182560215543708673587, −5.70393353877030362855966441639, −3.61361412423275293026603192528, −2.93329524727049682992831078618, −2.16510977081162991618350910538, −1.40570036687269561729357193042, 0, 0, 1.40570036687269561729357193042, 2.16510977081162991618350910538, 2.93329524727049682992831078618, 3.61361412423275293026603192528, 5.70393353877030362855966441639, 5.80957635182560215543708673587, 7.47970327921472559890882513517, 7.58890240033698295300761005100, 8.975210981503701117147819781525, 9.413814330920998732272832278168, 9.966871464894000697832937261643, 11.03093205288358349022703629731, 11.95371952249700270094874721146, 12.88315956060479463436269766843, 14.21704650252904920939280029947, 14.56332069409854110068722960153, 16.07009390458362862232076917801, 16.32505129086880965855091484654