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Properties

Label 4-2e12-1.1-c23e2-0-0
Degree $4$
Conductor $4096$
Sign $1$
Analytic cond. $46023.3$
Root an. cond. $14.6468$
Motivic weight $23$
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  − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4096\)    =    \(2^{12}\)
Sign: $1$
Analytic conductor: \(46023.3\)
Root analytic conductor: \(14.6468\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4096,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(0.2526120818\)
\(L(\frac12)\) \(\approx\) \(0.2526120818\)
\(L(\frac{25}{2})\) 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

−10.93314286502785279700492270371, −10.44833606175557248208106148941, −9.879913566658290617061240129422, −9.667720308598270535311391378986, −8.570983092570713224599106489979, −8.181382872519304270615515420223, −7.53467782161902787436728518598, −7.44584395615431310426400324124, −6.21821882584695292335682690381, −6.10060066659180529612633697685, −5.37519190743797491886878230082, −5.27736610147200845641926949336, −4.27940824675953138290896803481, −3.95782527980394037296406294396, −3.09079121385167159095496484487, −2.76788232808334818991637966875, −2.06163609819058944953195329483, −1.30376338176067444772754165917, −0.57732197670558260913795131502, −0.16543310342462936551917356945, 0.16543310342462936551917356945, 0.57732197670558260913795131502, 1.30376338176067444772754165917, 2.06163609819058944953195329483, 2.76788232808334818991637966875, 3.09079121385167159095496484487, 3.95782527980394037296406294396, 4.27940824675953138290896803481, 5.27736610147200845641926949336, 5.37519190743797491886878230082, 6.10060066659180529612633697685, 6.21821882584695292335682690381, 7.44584395615431310426400324124, 7.53467782161902787436728518598, 8.181382872519304270615515420223, 8.570983092570713224599106489979, 9.667720308598270535311391378986, 9.879913566658290617061240129422, 10.44833606175557248208106148941, 10.93314286502785279700492270371

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