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Properties

Label 4-2e8-1.1-c27e2-0-1
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $5460.75$
Root an. cond. $8.59633$
Motivic weight $27$
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  + 1.28e6·3-s + 5.44e9·5-s + 1.75e11·7-s − 6.18e12·9-s − 1.38e14·11-s − 7.53e14·13-s + 7.00e15·15-s − 2.97e16·17-s − 4.04e17·19-s + 2.25e17·21-s − 2.92e18·23-s + 1.19e19·25-s − 8.21e18·27-s − 1.55e19·29-s − 2.85e19·31-s − 1.77e20·33-s + 9.54e20·35-s + 1.86e21·37-s − 9.69e20·39-s + 9.08e21·41-s − 5.14e21·43-s − 3.36e22·45-s + 1.15e21·47-s − 9.64e22·49-s − 3.82e22·51-s + 1.06e23·53-s − 7.52e23·55-s + ⋯
L(s)  = 1  + 0.465·3-s + 1.99·5-s + 0.684·7-s − 0.810·9-s − 1.20·11-s − 0.689·13-s + 0.928·15-s − 0.728·17-s − 2.20·19-s + 0.318·21-s − 1.21·23-s + 1.60·25-s − 0.390·27-s − 0.281·29-s − 0.209·31-s − 0.562·33-s + 1.36·35-s + 1.26·37-s − 0.321·39-s + 1.53·41-s − 0.456·43-s − 1.61·45-s + 0.0307·47-s − 1.46·49-s − 0.339·51-s + 0.564·53-s − 2.40·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(5460.75\)
Root analytic conductor: \(8.59633\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 256,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{29}{2})\) 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$D_{4}$ \( 1 - 15880 p^{4} T + 1194221510 p^{8} T^{2} - 15880 p^{31} T^{3} + p^{54} T^{4} \)
5$D_{4}$ \( 1 - 217743516 p^{2} T + 5666334661152134 p^{5} T^{2} - 217743516 p^{29} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 - 25055994800 p T + 52976088888041064750 p^{4} T^{2} - 25055994800 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 + 12560667062904 p T + \)\(90\!\cdots\!46\)\( p^{3} T^{2} + 12560667062904 p^{28} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 57956446251620 p T + \)\(14\!\cdots\!30\)\( p^{2} T^{2} + 57956446251620 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 29753620331011740 T + \)\(20\!\cdots\!70\)\( p T^{2} + 29753620331011740 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 + 21292937388036040 p T + \)\(29\!\cdots\!98\)\( p^{2} T^{2} + 21292937388036040 p^{28} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 127351257527009520 p T + \)\(22\!\cdots\!70\)\( p^{2} T^{2} + 127351257527009520 p^{28} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 15546679995448558260 T + \)\(61\!\cdots\!18\)\( T^{2} + 15546679995448558260 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 + 28544554594467385024 T + \)\(31\!\cdots\!66\)\( T^{2} + 28544554594467385024 p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(49\!\cdots\!70\)\( T^{2} - \)\(18\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 - \)\(90\!\cdots\!64\)\( T + \)\(89\!\cdots\!86\)\( T^{2} - \)\(90\!\cdots\!64\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 + \)\(51\!\cdots\!00\)\( T + \)\(17\!\cdots\!50\)\( T^{2} + \)\(51\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 - \)\(11\!\cdots\!60\)\( T + \)\(27\!\cdots\!10\)\( T^{2} - \)\(11\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(63\!\cdots\!10\)\( T^{2} - \)\(10\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 + \)\(20\!\cdots\!80\)\( T + \)\(23\!\cdots\!38\)\( T^{2} + \)\(20\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(14\!\cdots\!44\)\( T + \)\(12\!\cdots\!26\)\( T^{2} - \)\(14\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 + \)\(30\!\cdots\!60\)\( T + \)\(20\!\cdots\!90\)\( T^{2} + \)\(30\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 - \)\(13\!\cdots\!16\)\( T + \)\(13\!\cdots\!46\)\( T^{2} - \)\(13\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 - \)\(52\!\cdots\!60\)\( T + \)\(30\!\cdots\!30\)\( T^{2} - \)\(52\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 + \)\(62\!\cdots\!40\)\( T + \)\(41\!\cdots\!18\)\( T^{2} + \)\(62\!\cdots\!40\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 + \)\(17\!\cdots\!80\)\( T + \)\(20\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 + \)\(31\!\cdots\!80\)\( T + \)\(11\!\cdots\!58\)\( T^{2} + \)\(31\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(65\!\cdots\!60\)\( T + \)\(48\!\cdots\!10\)\( T^{2} + \)\(65\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−12.81026536306242333613062994457, −12.52104615290299155280654742575, −11.05329488214768924124843278767, −11.00600309473511291469547010673, −9.944345959183639649049576047529, −9.714879840167727921664196191691, −8.808678805966874035475858653968, −8.294459797670413430011403394513, −7.68926962018852760348562612210, −6.59817148578998491440115772529, −5.91623673508132515856828743659, −5.65022518033751787079320178489, −4.72042739279402285827651574816, −4.19368178607859478404349590145, −2.65928944379303755776628808998, −2.61799227694409812018826469353, −1.94153131417606702139817818409, −1.51220891613028011845661210947, 0, 0, 1.51220891613028011845661210947, 1.94153131417606702139817818409, 2.61799227694409812018826469353, 2.65928944379303755776628808998, 4.19368178607859478404349590145, 4.72042739279402285827651574816, 5.65022518033751787079320178489, 5.91623673508132515856828743659, 6.59817148578998491440115772529, 7.68926962018852760348562612210, 8.294459797670413430011403394513, 8.808678805966874035475858653968, 9.714879840167727921664196191691, 9.944345959183639649049576047529, 11.00600309473511291469547010673, 11.05329488214768924124843278767, 12.52104615290299155280654742575, 12.81026536306242333613062994457

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