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Properties

Label 4-2e8-1.1-c45e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $42110.8$
Root an. cond. $14.3251$
Motivic weight $45$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.98e10·3-s + 4.39e15·5-s − 7.90e18·7-s − 2.69e20·9-s − 1.93e23·11-s − 2.31e24·13-s − 2.63e26·15-s − 6.57e27·17-s + 9.00e28·19-s + 4.73e29·21-s − 6.99e30·23-s + 6.00e30·25-s + 6.99e31·27-s + 2.21e33·29-s − 9.65e33·31-s + 1.15e34·33-s − 3.47e34·35-s − 1.91e35·37-s + 1.38e35·39-s − 1.85e36·41-s + 8.04e35·43-s − 1.18e36·45-s − 3.61e37·47-s − 1.64e38·49-s + 3.93e38·51-s − 6.66e38·53-s − 8.50e38·55-s + ⋯
L(s)  = 1  − 1.10·3-s + 0.824·5-s − 0.763·7-s − 0.0912·9-s − 0.716·11-s − 0.199·13-s − 0.908·15-s − 1.35·17-s + 1.52·19-s + 0.841·21-s − 1.60·23-s + 0.211·25-s + 0.435·27-s + 2.76·29-s − 2.68·31-s + 0.788·33-s − 0.630·35-s − 0.992·37-s + 0.219·39-s − 0.958·41-s + 0.141·43-s − 0.0752·45-s − 0.861·47-s − 1.53·49-s + 1.49·51-s − 1.06·53-s − 0.590·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(46-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+45/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(42110.8\)
Root analytic conductor: \(14.3251\)
Motivic weight: \(45\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :45/2, 45/2),\ 1)\)

Particular Values

\(L(23)\) \(\approx\) \(0.9547160232\)
\(L(\frac12)\) \(\approx\) \(0.9547160232\)
\(L(\frac{47}{2})\) 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$D_{4}$ \( 1 + 19953739064 p T + 195744721333750994 p^{9} T^{2} + 19953739064 p^{46} T^{3} + p^{90} T^{4} \)
5$D_{4}$ \( 1 - 175884559656204 p^{2} T + \)\(34\!\cdots\!22\)\( p^{8} T^{2} - 175884559656204 p^{47} T^{3} + p^{90} T^{4} \)
7$D_{4}$ \( 1 + 1128927599067328432 p T + \)\(13\!\cdots\!94\)\( p^{5} T^{2} + 1128927599067328432 p^{46} T^{3} + p^{90} T^{4} \)
11$D_{4}$ \( 1 + \)\(17\!\cdots\!04\)\( p T + \)\(10\!\cdots\!06\)\( p^{3} T^{2} + \)\(17\!\cdots\!04\)\( p^{46} T^{3} + p^{90} T^{4} \)
13$D_{4}$ \( 1 + \)\(17\!\cdots\!56\)\( p T + \)\(35\!\cdots\!62\)\( p^{4} T^{2} + \)\(17\!\cdots\!56\)\( p^{46} T^{3} + p^{90} T^{4} \)
17$D_{4}$ \( 1 + \)\(65\!\cdots\!36\)\( T + \)\(11\!\cdots\!42\)\( p^{2} T^{2} + \)\(65\!\cdots\!36\)\( p^{45} T^{3} + p^{90} T^{4} \)
19$D_{4}$ \( 1 - \)\(90\!\cdots\!20\)\( T + \)\(19\!\cdots\!18\)\( p^{2} T^{2} - \)\(90\!\cdots\!20\)\( p^{45} T^{3} + p^{90} T^{4} \)
23$D_{4}$ \( 1 + \)\(30\!\cdots\!44\)\( p T + \)\(72\!\cdots\!18\)\( p^{2} T^{2} + \)\(30\!\cdots\!44\)\( p^{46} T^{3} + p^{90} T^{4} \)
29$D_{4}$ \( 1 - \)\(76\!\cdots\!60\)\( p T + \)\(29\!\cdots\!78\)\( p^{2} T^{2} - \)\(76\!\cdots\!60\)\( p^{46} T^{3} + p^{90} T^{4} \)
31$D_{4}$ \( 1 + \)\(31\!\cdots\!64\)\( p T + \)\(48\!\cdots\!06\)\( p^{2} T^{2} + \)\(31\!\cdots\!64\)\( p^{46} T^{3} + p^{90} T^{4} \)
37$D_{4}$ \( 1 + \)\(51\!\cdots\!08\)\( p T + \)\(60\!\cdots\!22\)\( p^{2} T^{2} + \)\(51\!\cdots\!08\)\( p^{46} T^{3} + p^{90} T^{4} \)
41$D_{4}$ \( 1 + \)\(45\!\cdots\!36\)\( p T + \)\(37\!\cdots\!66\)\( p^{2} T^{2} + \)\(45\!\cdots\!36\)\( p^{46} T^{3} + p^{90} T^{4} \)
43$D_{4}$ \( 1 - \)\(18\!\cdots\!96\)\( p T + \)\(19\!\cdots\!18\)\( p^{2} T^{2} - \)\(18\!\cdots\!96\)\( p^{46} T^{3} + p^{90} T^{4} \)
47$D_{4}$ \( 1 + \)\(36\!\cdots\!04\)\( T + \)\(38\!\cdots\!18\)\( T^{2} + \)\(36\!\cdots\!04\)\( p^{45} T^{3} + p^{90} T^{4} \)
53$D_{4}$ \( 1 + \)\(66\!\cdots\!88\)\( T + \)\(20\!\cdots\!22\)\( T^{2} + \)\(66\!\cdots\!88\)\( p^{45} T^{3} + p^{90} T^{4} \)
59$D_{4}$ \( 1 - \)\(14\!\cdots\!20\)\( T + \)\(53\!\cdots\!98\)\( T^{2} - \)\(14\!\cdots\!20\)\( p^{45} T^{3} + p^{90} T^{4} \)
61$D_{4}$ \( 1 - \)\(16\!\cdots\!64\)\( T + \)\(42\!\cdots\!26\)\( T^{2} - \)\(16\!\cdots\!64\)\( p^{45} T^{3} + p^{90} T^{4} \)
67$D_{4}$ \( 1 - \)\(82\!\cdots\!76\)\( T + \)\(22\!\cdots\!58\)\( T^{2} - \)\(82\!\cdots\!76\)\( p^{45} T^{3} + p^{90} T^{4} \)
71$D_{4}$ \( 1 + \)\(11\!\cdots\!64\)\( T + \)\(23\!\cdots\!26\)\( T^{2} + \)\(11\!\cdots\!64\)\( p^{45} T^{3} + p^{90} T^{4} \)
73$D_{4}$ \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(15\!\cdots\!42\)\( T^{2} - \)\(13\!\cdots\!32\)\( p^{45} T^{3} + p^{90} T^{4} \)
79$D_{4}$ \( 1 + \)\(33\!\cdots\!00\)\( T + \)\(51\!\cdots\!98\)\( T^{2} + \)\(33\!\cdots\!00\)\( p^{45} T^{3} + p^{90} T^{4} \)
83$D_{4}$ \( 1 - \)\(95\!\cdots\!48\)\( T + \)\(32\!\cdots\!62\)\( T^{2} - \)\(95\!\cdots\!48\)\( p^{45} T^{3} + p^{90} T^{4} \)
89$D_{4}$ \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(67\!\cdots\!80\)\( p^{45} T^{3} + p^{90} T^{4} \)
97$D_{4}$ \( 1 + \)\(82\!\cdots\!36\)\( T + \)\(67\!\cdots\!38\)\( T^{2} + \)\(82\!\cdots\!36\)\( p^{45} T^{3} + p^{90} T^{4} \)
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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

−11.12066670862156962048263848946, −10.97920734730021811762961210850, −10.02285748074062862855386230227, −9.892882769318761033202242416229, −9.223935526755283020561020113596, −8.473209690879826637876524436964, −7.929646710038069030629410268148, −7.06360051898744031914855732906, −6.48265664573973357366887100452, −6.28501907350612771589118112821, −5.38761405385321651168613611139, −5.30987760009749600715101804239, −4.69373697251674390557607629252, −3.78234011704426303123459032681, −3.18497977807680085892662106111, −2.67121174089293329786132299390, −1.89294187109402124902556608836, −1.62917407215985596990576288962, −0.53471218798620359275210720941, −0.32327180152169082607259351645, 0.32327180152169082607259351645, 0.53471218798620359275210720941, 1.62917407215985596990576288962, 1.89294187109402124902556608836, 2.67121174089293329786132299390, 3.18497977807680085892662106111, 3.78234011704426303123459032681, 4.69373697251674390557607629252, 5.30987760009749600715101804239, 5.38761405385321651168613611139, 6.28501907350612771589118112821, 6.48265664573973357366887100452, 7.06360051898744031914855732906, 7.929646710038069030629410268148, 8.473209690879826637876524436964, 9.223935526755283020561020113596, 9.892882769318761033202242416229, 10.02285748074062862855386230227, 10.97920734730021811762961210850, 11.12066670862156962048263848946

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