Dirichlet series
| L(s) = 1 | − 1.22e11·3-s + 1.81e16·5-s − 1.61e20·7-s + 2.06e21·9-s − 1.14e24·11-s − 9.11e25·13-s − 2.22e27·15-s − 1.02e29·17-s − 1.02e30·19-s + 1.96e31·21-s − 8.76e30·23-s − 7.99e32·25-s − 1.92e33·27-s + 4.48e34·29-s − 2.07e35·31-s + 1.40e35·33-s − 2.92e36·35-s + 1.54e37·37-s + 1.11e37·39-s − 1.83e38·41-s − 2.51e38·43-s + 3.74e37·45-s − 2.87e39·47-s + 9.76e39·49-s + 1.25e40·51-s + 3.62e40·53-s − 2.08e40·55-s + ⋯ |
| L(s) = 1 | − 0.749·3-s + 0.681·5-s − 2.22·7-s + 0.0775·9-s − 0.386·11-s − 0.605·13-s − 0.511·15-s − 1.24·17-s − 0.908·19-s + 1.66·21-s − 0.0874·23-s − 1.12·25-s − 0.444·27-s + 1.93·29-s − 1.86·31-s + 0.290·33-s − 1.51·35-s + 2.16·37-s + 0.454·39-s − 2.30·41-s − 1.03·43-s + 0.0528·45-s − 1.46·47-s + 1.86·49-s + 0.936·51-s + 1.09·53-s − 0.263·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+47/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(256\) = \(2^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(50109.8\) |
| Root analytic conductor: | \(14.9616\) |
| Motivic weight: | \(47\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(2\) |
| Selberg data: | \((4,\ 256,\ (\ :47/2, 47/2),\ 1)\) |
Particular Values
| \(L(24)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{49}{2})\) | 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 | \( 1 \) | |
| good | 3 | $D_{4}$ | \( 1 + 4529253512 p^{3} T + 218342028119393782 p^{10} T^{2} + 4529253512 p^{50} T^{3} + p^{94} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3635699567838828 p T + \)\(14\!\cdots\!54\)\( p^{7} T^{2} - 3635699567838828 p^{48} T^{3} + p^{94} T^{4} \) | |
| 7 | $D_{4}$ | \( 1 + 3286092764408710768 p^{2} T + \)\(19\!\cdots\!94\)\( p^{7} T^{2} + 3286092764408710768 p^{49} T^{3} + p^{94} T^{4} \) | |
| 11 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!84\)\( T + \)\(50\!\cdots\!26\)\( p^{3} T^{2} + \)\(11\!\cdots\!84\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 13 | $D_{4}$ | \( 1 + \)\(70\!\cdots\!72\)\( p T + \)\(42\!\cdots\!78\)\( p^{4} T^{2} + \)\(70\!\cdots\!72\)\( p^{48} T^{3} + p^{94} T^{4} \) | |
| 17 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(45\!\cdots\!46\)\( p T^{2} + \)\(10\!\cdots\!88\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 19 | $D_{4}$ | \( 1 + \)\(53\!\cdots\!80\)\( p T + \)\(34\!\cdots\!42\)\( p^{3} T^{2} + \)\(53\!\cdots\!80\)\( p^{48} T^{3} + p^{94} T^{4} \) | |
| 23 | $D_{4}$ | \( 1 + \)\(16\!\cdots\!56\)\( p^{2} T + \)\(36\!\cdots\!22\)\( p^{2} T^{2} + \)\(16\!\cdots\!56\)\( p^{49} T^{3} + p^{94} T^{4} \) | |
| 29 | $D_{4}$ | \( 1 - \)\(44\!\cdots\!60\)\( T + \)\(13\!\cdots\!98\)\( p^{2} T^{2} - \)\(44\!\cdots\!60\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 31 | $D_{4}$ | \( 1 + \)\(66\!\cdots\!84\)\( p T + \)\(32\!\cdots\!66\)\( p^{2} T^{2} + \)\(66\!\cdots\!84\)\( p^{48} T^{3} + p^{94} T^{4} \) | |
| 37 | $D_{4}$ | \( 1 - \)\(41\!\cdots\!96\)\( p T + \)\(11\!\cdots\!18\)\( p^{2} T^{2} - \)\(41\!\cdots\!96\)\( p^{48} T^{3} + p^{94} T^{4} \) | |
| 41 | $D_{4}$ | \( 1 + \)\(18\!\cdots\!16\)\( T + \)\(51\!\cdots\!86\)\( p T^{2} + \)\(18\!\cdots\!16\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 43 | $D_{4}$ | \( 1 + \)\(58\!\cdots\!28\)\( p T + \)\(40\!\cdots\!82\)\( p^{2} T^{2} + \)\(58\!\cdots\!28\)\( p^{48} T^{3} + p^{94} T^{4} \) | |
| 47 | $D_{4}$ | \( 1 + \)\(28\!\cdots\!12\)\( T + \)\(55\!\cdots\!62\)\( T^{2} + \)\(28\!\cdots\!12\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 53 | $D_{4}$ | \( 1 - \)\(36\!\cdots\!64\)\( T + \)\(24\!\cdots\!98\)\( T^{2} - \)\(36\!\cdots\!64\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 59 | $D_{4}$ | \( 1 + \)\(21\!\cdots\!20\)\( T + \)\(11\!\cdots\!38\)\( T^{2} + \)\(21\!\cdots\!20\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 61 | $D_{4}$ | \( 1 - \)\(23\!\cdots\!04\)\( T + \)\(29\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!04\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 67 | $D_{4}$ | \( 1 + \)\(96\!\cdots\!92\)\( T + \)\(11\!\cdots\!62\)\( T^{2} + \)\(96\!\cdots\!92\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 71 | $D_{4}$ | \( 1 - \)\(34\!\cdots\!56\)\( T + \)\(23\!\cdots\!66\)\( T^{2} - \)\(34\!\cdots\!56\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 73 | $D_{4}$ | \( 1 + \)\(44\!\cdots\!36\)\( T + \)\(77\!\cdots\!18\)\( T^{2} + \)\(44\!\cdots\!36\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 79 | $D_{4}$ | \( 1 + \)\(45\!\cdots\!00\)\( T + \)\(19\!\cdots\!18\)\( T^{2} + \)\(45\!\cdots\!00\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 83 | $D_{4}$ | \( 1 - \)\(13\!\cdots\!16\)\( T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(13\!\cdots\!16\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 89 | $D_{4}$ | \( 1 - \)\(45\!\cdots\!20\)\( T + \)\(87\!\cdots\!58\)\( T^{2} - \)\(45\!\cdots\!20\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| 97 | $D_{4}$ | \( 1 - \)\(53\!\cdots\!92\)\( T + \)\(44\!\cdots\!42\)\( T^{2} - \)\(53\!\cdots\!92\)\( p^{47} T^{3} + p^{94} T^{4} \) | |
| show more | |||
| show less | |||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.27622059228305360595544981590, −10.07789832289830805710751787223, −9.749522975651381259976972263179, −9.027013306239963825896792175591, −8.508991997693947795728229181152, −7.66678576549329135752121229094, −6.78462930171348761178611017645, −6.62189769019205620532676133246, −6.13942004989544240637422613737, −5.69064911010242966154825839288, −4.98252134289198479240623560697, −4.45126071576011647936890878764, −3.61018050310744334016212810834, −3.30225381955442147634202318207, −2.34528673507094922818275592083, −2.31362797628089112412045546328, −1.47805944243325553993454076501, −0.54146628352551711313251747693, 0, 0, 0.54146628352551711313251747693, 1.47805944243325553993454076501, 2.31362797628089112412045546328, 2.34528673507094922818275592083, 3.30225381955442147634202318207, 3.61018050310744334016212810834, 4.45126071576011647936890878764, 4.98252134289198479240623560697, 5.69064911010242966154825839288, 6.13942004989544240637422613737, 6.62189769019205620532676133246, 6.78462930171348761178611017645, 7.66678576549329135752121229094, 8.508991997693947795728229181152, 9.027013306239963825896792175591, 9.749522975651381259976972263179, 10.07789832289830805710751787223, 10.27622059228305360595544981590