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Properties

Label 4-2e8-1.1-c31e2-0-2
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $9487.42$
Root an. cond. $9.86931$
Motivic weight $31$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.67e7·3-s − 2.67e10·5-s + 2.35e13·7-s − 1.15e14·9-s − 2.51e16·11-s + 1.89e17·13-s + 4.47e17·15-s + 9.20e17·17-s + 1.00e20·19-s − 3.92e20·21-s − 5.10e20·23-s + 4.22e21·25-s − 1.78e21·27-s − 3.93e22·29-s − 4.97e22·31-s + 4.20e23·33-s − 6.29e23·35-s + 1.88e24·37-s − 3.16e24·39-s − 2.92e25·41-s + 2.90e25·43-s + 3.09e24·45-s − 1.33e26·47-s + 1.00e26·49-s − 1.53e25·51-s − 4.56e26·53-s + 6.73e26·55-s + ⋯
L(s)  = 1  − 0.672·3-s − 0.392·5-s + 1.87·7-s − 0.187·9-s − 1.81·11-s + 1.02·13-s + 0.263·15-s + 0.0780·17-s + 1.51·19-s − 1.25·21-s − 0.399·23-s + 0.907·25-s − 0.116·27-s − 0.847·29-s − 0.380·31-s + 1.22·33-s − 0.733·35-s + 0.927·37-s − 0.690·39-s − 2.94·41-s + 1.39·43-s + 0.0734·45-s − 1.61·47-s + 0.634·49-s − 0.0524·51-s − 0.856·53-s + 0.711·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(9487.42\)
Root analytic conductor: \(9.86931\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 256,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{33}{2})\) 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)$
bad2 \( 1 \)
good3$D_{4}$ \( 1 + 5572168 p T + 180667888754 p^{7} T^{2} + 5572168 p^{32} T^{3} + p^{62} T^{4} \)
5$D_{4}$ \( 1 + 1070522628 p^{2} T - 44945775705560938 p^{7} T^{2} + 1070522628 p^{33} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 - 23503702110608 T + \)\(13\!\cdots\!14\)\( p^{3} T^{2} - 23503702110608 p^{31} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 2286467043501144 p T + \)\(37\!\cdots\!06\)\( p^{3} T^{2} + 2286467043501144 p^{32} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 - 189324344559709324 T + \)\(33\!\cdots\!22\)\( p^{2} T^{2} - 189324344559709324 p^{31} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 - 920816829890675172 T + \)\(16\!\cdots\!86\)\( p T^{2} - 920816829890675172 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(55\!\cdots\!02\)\( p T^{2} - \)\(10\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 + 22195146899293731888 p T + \)\(64\!\cdots\!62\)\( p^{2} T^{2} + 22195146899293731888 p^{32} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 + 46811908508319393780 p^{2} T + \)\(46\!\cdots\!38\)\( p^{2} T^{2} + 46811908508319393780 p^{33} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 + \)\(49\!\cdots\!84\)\( T + \)\(25\!\cdots\!26\)\( T^{2} + \)\(49\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 - \)\(18\!\cdots\!52\)\( T + \)\(67\!\cdots\!02\)\( T^{2} - \)\(18\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 + \)\(29\!\cdots\!36\)\( T + \)\(40\!\cdots\!06\)\( T^{2} + \)\(29\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(29\!\cdots\!36\)\( T + \)\(92\!\cdots\!38\)\( T^{2} - \)\(29\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 + \)\(13\!\cdots\!52\)\( T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(13\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 + \)\(45\!\cdots\!36\)\( T + \)\(36\!\cdots\!18\)\( T^{2} + \)\(45\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 - \)\(25\!\cdots\!60\)\( T + \)\(17\!\cdots\!18\)\( T^{2} - \)\(25\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 - \)\(73\!\cdots\!64\)\( T + \)\(43\!\cdots\!46\)\( T^{2} - \)\(73\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(21\!\cdots\!92\)\( T + \)\(83\!\cdots\!82\)\( T^{2} + \)\(21\!\cdots\!92\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 + \)\(36\!\cdots\!84\)\( T + \)\(32\!\cdots\!06\)\( T^{2} + \)\(36\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 - \)\(16\!\cdots\!84\)\( T + \)\(81\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 - \)\(43\!\cdots\!20\)\( T + \)\(89\!\cdots\!58\)\( T^{2} - \)\(43\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 + \)\(41\!\cdots\!24\)\( T + \)\(61\!\cdots\!78\)\( T^{2} + \)\(41\!\cdots\!24\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 - \)\(52\!\cdots\!20\)\( T + \)\(54\!\cdots\!78\)\( T^{2} - \)\(52\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(93\!\cdots\!72\)\( T + \)\(92\!\cdots\!02\)\( T^{2} - \)\(93\!\cdots\!72\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−11.76900527025586407974483824618, −11.65561307442662043562336298490, −11.02162546959111316403470816720, −10.63326718417425522541641616671, −9.863006891657828491127173914200, −8.878901270729683995918936173034, −8.113832623867508829072573368136, −7.916872647164126035725101954444, −7.30750712667681037105829952956, −6.31311994384657614277237839718, −5.45860834439378651064352176155, −5.17156202082758036454477162869, −4.76060726007105851735180655159, −3.78452936837964503573834545363, −3.09903596449593701648624879719, −2.35211121276724249993735711164, −1.35698654493786156034835424023, −1.30873956317347105844104177374, 0, 0, 1.30873956317347105844104177374, 1.35698654493786156034835424023, 2.35211121276724249993735711164, 3.09903596449593701648624879719, 3.78452936837964503573834545363, 4.76060726007105851735180655159, 5.17156202082758036454477162869, 5.45860834439378651064352176155, 6.31311994384657614277237839718, 7.30750712667681037105829952956, 7.916872647164126035725101954444, 8.113832623867508829072573368136, 8.878901270729683995918936173034, 9.863006891657828491127173914200, 10.63326718417425522541641616671, 11.02162546959111316403470816720, 11.65561307442662043562336298490, 11.76900527025586407974483824618

Graph of the $Z$-function along the critical line