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Properties

Label 4-2e8-1.1-c23e2-0-1
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $2876.46$
Root an. cond. $7.32343$
Motivic weight $23$
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  − 3.39e5·3-s + 7.30e7·5-s + 1.35e9·7-s − 5.40e10·9-s − 8.56e11·11-s + 4.37e12·13-s − 2.48e13·15-s + 2.54e14·17-s − 4.26e12·19-s − 4.61e14·21-s + 8.14e15·23-s − 1.51e16·25-s + 4.38e16·27-s + 2.08e16·29-s − 1.37e17·31-s + 2.90e17·33-s + 9.93e16·35-s − 8.97e17·37-s − 1.48e18·39-s − 2.29e18·41-s + 1.75e18·43-s − 3.94e18·45-s − 1.57e19·47-s − 3.31e19·49-s − 8.62e19·51-s − 1.40e20·53-s − 6.26e19·55-s + ⋯
L(s)  = 1  − 1.10·3-s + 0.669·5-s + 0.259·7-s − 0.573·9-s − 0.905·11-s + 0.677·13-s − 0.740·15-s + 1.79·17-s − 0.00839·19-s − 0.287·21-s + 1.78·23-s − 1.27·25-s + 1.51·27-s + 0.316·29-s − 0.973·31-s + 1.00·33-s + 0.173·35-s − 0.829·37-s − 0.749·39-s − 0.651·41-s + 0.287·43-s − 0.384·45-s − 0.929·47-s − 1.21·49-s − 1.98·51-s − 2.07·53-s − 0.605·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(2876.46\)
Root analytic conductor: \(7.32343\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 256,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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$D_{4}$ \( 1 + 37720 p^{2} T + 77396530 p^{7} T^{2} + 37720 p^{25} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 - 14613804 p T + 32768971378174 p^{4} T^{2} - 14613804 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 194169200 p T + 102109123349477250 p^{3} T^{2} - 194169200 p^{24} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 77891088024 p T + \)\(16\!\cdots\!66\)\( p^{2} T^{2} + 77891088024 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 4376109322060 T + \)\(42\!\cdots\!90\)\( p T^{2} - 4376109322060 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 - 14942832211620 p T + \)\(15\!\cdots\!10\)\( p^{2} T^{2} - 14942832211620 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 224242156840 p T + \)\(34\!\cdots\!38\)\( p^{2} T^{2} + 224242156840 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 - 8144713079008560 T + \)\(58\!\cdots\!30\)\( T^{2} - 8144713079008560 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 20818433601623340 T + \)\(77\!\cdots\!78\)\( T^{2} - 20818433601623340 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 137714017177000384 T + \)\(40\!\cdots\!46\)\( T^{2} + 137714017177000384 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 897721264408967780 T + \)\(19\!\cdots\!70\)\( T^{2} + 897721264408967780 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 + 2294435477168314956 T + \)\(19\!\cdots\!26\)\( T^{2} + 2294435477168314956 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 - 1750760768619855800 T + \)\(73\!\cdots\!50\)\( T^{2} - 1750760768619855800 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 15759744217656780960 T + \)\(36\!\cdots\!10\)\( T^{2} + 15759744217656780960 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(13\!\cdots\!10\)\( T^{2} + \)\(14\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 + \)\(28\!\cdots\!80\)\( T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(28\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(18\!\cdots\!36\)\( T + \)\(96\!\cdots\!86\)\( T^{2} + \)\(18\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(17\!\cdots\!40\)\( T + \)\(24\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 + \)\(30\!\cdots\!24\)\( T + \)\(93\!\cdots\!66\)\( T^{2} + \)\(30\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(80\!\cdots\!60\)\( T + \)\(30\!\cdots\!30\)\( T^{2} + \)\(80\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + \)\(62\!\cdots\!40\)\( T + \)\(47\!\cdots\!78\)\( T^{2} + \)\(62\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 + \)\(68\!\cdots\!20\)\( T + \)\(16\!\cdots\!90\)\( T^{2} + \)\(68\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(63\!\cdots\!20\)\( T + \)\(13\!\cdots\!38\)\( T^{2} - \)\(63\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 + \)\(31\!\cdots\!40\)\( T + \)\(53\!\cdots\!10\)\( T^{2} + \)\(31\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−13.23579620688203032999677493489, −12.90706692586499688421606331728, −11.84210463950351238458279361050, −11.59474244691236034124576973414, −10.70841091094538958107613104905, −10.39819033798327568298290185448, −9.487130186096180434978851802582, −8.737116981779880356042336432246, −7.924193783814898844605917553340, −7.26316909152946154236682831623, −6.08591496381759003022944704681, −5.87434013708782788777677155496, −5.22621896371540424009009089828, −4.67426436606585698641478957256, −3.12001488390327276866364658945, −3.09980718338561759622781009122, −1.61385002027622331233153944385, −1.31313979235274195425059904199, 0, 0, 1.31313979235274195425059904199, 1.61385002027622331233153944385, 3.09980718338561759622781009122, 3.12001488390327276866364658945, 4.67426436606585698641478957256, 5.22621896371540424009009089828, 5.87434013708782788777677155496, 6.08591496381759003022944704681, 7.26316909152946154236682831623, 7.924193783814898844605917553340, 8.737116981779880356042336432246, 9.487130186096180434978851802582, 10.39819033798327568298290185448, 10.70841091094538958107613104905, 11.59474244691236034124576973414, 11.84210463950351238458279361050, 12.90706692586499688421606331728, 13.23579620688203032999677493489

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