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Properties

Label 4-2e8-1.1-c23e2-0-0
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  − 1.70e5·3-s − 9.22e7·5-s − 1.92e8·7-s − 7.53e10·9-s + 1.24e12·11-s − 7.46e12·13-s + 1.57e13·15-s − 3.74e14·17-s + 8.40e14·19-s + 3.27e13·21-s − 6.43e15·23-s + 9.12e15·25-s + 1.45e16·27-s − 6.80e16·29-s + 1.45e17·31-s − 2.12e17·33-s + 1.77e16·35-s + 1.21e18·37-s + 1.27e18·39-s + 5.03e18·41-s − 9.35e17·43-s + 6.94e18·45-s + 8.43e18·47-s − 5.43e19·49-s + 6.39e19·51-s + 3.76e18·53-s − 1.15e20·55-s + ⋯
L(s)  = 1  − 0.555·3-s − 0.845·5-s − 0.0367·7-s − 0.799·9-s + 1.31·11-s − 1.15·13-s + 0.469·15-s − 2.65·17-s + 1.65·19-s + 0.0204·21-s − 1.40·23-s + 0.765·25-s + 0.504·27-s − 1.03·29-s + 1.02·31-s − 0.732·33-s + 0.0310·35-s + 1.11·37-s + 0.641·39-s + 1.42·41-s − 0.153·43-s + 0.675·45-s + 0.497·47-s − 1.98·49-s + 1.47·51-s + 0.0557·53-s − 1.11·55-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$D_{4}$ \( 1 + 56840 p T + 429554626 p^{5} T^{2} + 56840 p^{24} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 + 18453204 p T - 985891951106 p^{4} T^{2} + 18453204 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 192083440 T + 1109027370312762510 p^{2} T^{2} + 192083440 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 - 113379517800 p T + \)\(11\!\cdots\!22\)\( p^{2} T^{2} - 113379517800 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 + 7460299980980 T + \)\(37\!\cdots\!86\)\( p T^{2} + 7460299980980 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 22052785170780 p T + \)\(25\!\cdots\!98\)\( p^{2} T^{2} + 22052785170780 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 - 44229488379608 p T + \)\(36\!\cdots\!26\)\( p T^{2} - 44229488379608 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 279711214046640 p T + \)\(40\!\cdots\!90\)\( T^{2} + 279711214046640 p^{24} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 + 68055499247434452 T + \)\(86\!\cdots\!54\)\( T^{2} + 68055499247434452 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 - 145584514546845248 T + \)\(81\!\cdots\!58\)\( T^{2} - 145584514546845248 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 - 1211894143551157660 T + \)\(25\!\cdots\!30\)\( T^{2} - 1211894143551157660 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 5036778367134688692 T + \)\(27\!\cdots\!58\)\( T^{2} - 5036778367134688692 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 935180945919935560 T + \)\(58\!\cdots\!14\)\( T^{2} + 935180945919935560 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 - 8431168896277596000 T + \)\(26\!\cdots\!22\)\( T^{2} - 8431168896277596000 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - 3763137197370204540 T + \)\(89\!\cdots\!30\)\( T^{2} - 3763137197370204540 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 + \)\(34\!\cdots\!16\)\( T + \)\(12\!\cdots\!22\)\( T^{2} + \)\(34\!\cdots\!16\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(58\!\cdots\!56\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(58\!\cdots\!56\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 + \)\(21\!\cdots\!60\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(21\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 + \)\(38\!\cdots\!96\)\( T + \)\(10\!\cdots\!26\)\( T^{2} + \)\(38\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(68\!\cdots\!80\)\( T + \)\(74\!\cdots\!18\)\( T^{2} + \)\(68\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - \)\(64\!\cdots\!36\)\( T + \)\(63\!\cdots\!02\)\( T^{2} - \)\(64\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(18\!\cdots\!00\)\( T + \)\(36\!\cdots\!70\)\( T^{2} - \)\(18\!\cdots\!00\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(50\!\cdots\!32\)\( T + \)\(95\!\cdots\!94\)\( T^{2} - \)\(50\!\cdots\!32\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(12\!\cdots\!80\)\( T + \)\(12\!\cdots\!10\)\( T^{2} - \)\(12\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−13.49078290504419887649352877590, −12.60676048594949882451455860715, −11.87669914225133017743554309197, −11.46917134217802926977223246844, −11.20270210525762399831014309826, −10.12826739535833006599247823623, −9.164102935271440150447245686353, −8.979262028732499319065425585508, −7.76269305983842904776616556016, −7.40026193934570824329902631597, −6.25683087375138814605145543035, −6.13805987047551784383092900603, −4.72051857840418627585621023531, −4.54876028617150041418762890647, −3.59268408982819381909128868743, −2.76277434277078194725047699844, −1.99406104092649059251857345118, −1.00915919513231014744326139757, 0, 0, 1.00915919513231014744326139757, 1.99406104092649059251857345118, 2.76277434277078194725047699844, 3.59268408982819381909128868743, 4.54876028617150041418762890647, 4.72051857840418627585621023531, 6.13805987047551784383092900603, 6.25683087375138814605145543035, 7.40026193934570824329902631597, 7.76269305983842904776616556016, 8.979262028732499319065425585508, 9.164102935271440150447245686353, 10.12826739535833006599247823623, 11.20270210525762399831014309826, 11.46917134217802926977223246844, 11.87669914225133017743554309197, 12.60676048594949882451455860715, 13.49078290504419887649352877590

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