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Properties

Label 4-2e12-1.1-c21e2-0-2
Degree $4$
Conductor $4096$
Sign $1$
Analytic cond. $31992.8$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.05e5·3-s − 2.10e6·5-s − 4.44e8·7-s + 6.33e8·9-s + 5.38e10·11-s + 4.90e11·13-s + 2.22e11·15-s − 6.59e12·17-s + 1.93e13·19-s + 4.68e13·21-s + 4.09e14·23-s − 9.45e14·25-s − 6.45e13·27-s + 2.40e15·29-s + 8.68e15·31-s − 5.67e15·33-s + 9.37e14·35-s + 2.18e15·37-s − 5.17e16·39-s + 6.81e16·41-s + 2.64e17·43-s − 1.33e15·45-s − 4.26e17·47-s − 7.33e17·49-s + 6.95e17·51-s + 3.05e18·53-s − 1.13e17·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.0965·5-s − 0.595·7-s + 0.0605·9-s + 0.625·11-s + 0.986·13-s + 0.0995·15-s − 0.793·17-s + 0.722·19-s + 0.613·21-s + 2.06·23-s − 1.98·25-s − 0.0603·27-s + 1.06·29-s + 1.90·31-s − 0.644·33-s + 0.0574·35-s + 0.0747·37-s − 1.01·39-s + 0.793·41-s + 1.86·43-s − 0.00584·45-s − 1.18·47-s − 1.31·49-s + 0.817·51-s + 2.40·53-s − 0.0603·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(31992.8\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4096,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(2.976537162\)
\(L(\frac12)\) \(\approx\) \(2.976537162\)
\(L(\frac{23}{2})\) 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$D_{4}$ \( 1 + 35144 p T + 129409046 p^{4} T^{2} + 35144 p^{22} T^{3} + p^{42} T^{4} \)
5$D_{4}$ \( 1 + 421628 p T + 7595986878694 p^{3} T^{2} + 421628 p^{22} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 + 444771792 T + 132991518054685058 p T^{2} + 444771792 p^{21} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 - 53806403320 T + 97941017461487903606 p^{2} T^{2} - 53806403320 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 - 490366676932 T + \)\(23\!\cdots\!06\)\( p T^{2} - 490366676932 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 + 387874392476 p T + \)\(47\!\cdots\!06\)\( p^{2} T^{2} + 387874392476 p^{22} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 - 1015915680280 p T - \)\(69\!\cdots\!18\)\( p^{2} T^{2} - 1015915680280 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 - 409737865776272 T + \)\(10\!\cdots\!18\)\( T^{2} - 409737865776272 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 - 2404787522145060 T + \)\(46\!\cdots\!74\)\( T^{2} - 2404787522145060 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 - 8689907170559168 T + \)\(56\!\cdots\!82\)\( T^{2} - 8689907170559168 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 - 2186204096251860 T + \)\(16\!\cdots\!58\)\( T^{2} - 2186204096251860 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 - 68178038573558676 T + \)\(15\!\cdots\!90\)\( T^{2} - 68178038573558676 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 - 264529652266004024 T + \)\(49\!\cdots\!34\)\( T^{2} - 264529652266004024 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 + 426494411558622432 T + \)\(23\!\cdots\!50\)\( T^{2} + 426494411558622432 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 - 3055980275589518132 T + \)\(54\!\cdots\!62\)\( T^{2} - 3055980275589518132 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 783424997522814424 T + \)\(30\!\cdots\!06\)\( T^{2} - 783424997522814424 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 - 7177279049078597092 T + \)\(67\!\cdots\!62\)\( T^{2} - 7177279049078597092 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 - 16674123174011538088 T + \)\(43\!\cdots\!06\)\( T^{2} - 16674123174011538088 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 + 9448263149848716368 T + \)\(82\!\cdots\!02\)\( T^{2} + 9448263149848716368 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 + 11586140334503007532 T - \)\(95\!\cdots\!02\)\( T^{2} + 11586140334503007532 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 - 85280702218715897824 T + \)\(14\!\cdots\!38\)\( T^{2} - 85280702218715897824 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 + \)\(38\!\cdots\!16\)\( T + \)\(71\!\cdots\!14\)\( T^{2} + \)\(38\!\cdots\!16\)\( p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 + 59742932430695979660 T + \)\(15\!\cdots\!22\)\( T^{2} + 59742932430695979660 p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 - \)\(78\!\cdots\!88\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(78\!\cdots\!88\)\( p^{21} T^{3} + p^{42} T^{4} \)
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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

−11.27937878089901599175919779127, −10.88597695593321023699550620730, −9.998434573626815520573343360860, −9.769773121567286165520878550247, −8.985130828071700695340072258055, −8.589845587635188152189073409972, −7.905381000936628771743230317440, −7.17515605261103101122261309253, −6.54636941188842671765589761886, −6.34673323524052930405137245176, −5.62514885301899606820011858812, −5.31080574529752454719461024020, −4.32896460018308725958145508280, −4.15389851138012206902918767382, −3.25845603216450371752625366986, −2.83318931769387687807282483514, −2.08051902969554027680736771574, −1.26462427732856764854179250856, −0.63742141272551716310797519248, −0.60015290648601372587670666729, 0.60015290648601372587670666729, 0.63742141272551716310797519248, 1.26462427732856764854179250856, 2.08051902969554027680736771574, 2.83318931769387687807282483514, 3.25845603216450371752625366986, 4.15389851138012206902918767382, 4.32896460018308725958145508280, 5.31080574529752454719461024020, 5.62514885301899606820011858812, 6.34673323524052930405137245176, 6.54636941188842671765589761886, 7.17515605261103101122261309253, 7.905381000936628771743230317440, 8.589845587635188152189073409972, 8.985130828071700695340072258055, 9.769773121567286165520878550247, 9.998434573626815520573343360860, 10.88597695593321023699550620730, 11.27937878089901599175919779127

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