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Properties

Label 4-2e12-1.1-c23e2-0-4
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 $2$

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

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

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: \(2\)
Selberg data: \((4,\ 4096,\ (\ :23/2, 23/2),\ 1)\)

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

−10.08531613270476264631197543356, −9.895293886851635573468593754445, −9.072644867017902083380598720847, −8.891119794484000977659631706985, −8.078431405365765655048705985109, −7.972344528036566289398435306604, −7.20076247862321241337333359340, −6.93015639305829947630650925503, −5.85667632587706229574287450487, −5.53019369799135014166152448687, −4.91247447846222624710035446709, −4.13672942642286505432777441969, −3.46953405887859205323107349635, −3.43328400577464420303536908336, −2.63736641430245155695974597026, −2.26029807067196141916295942649, −1.24201727822472133475307307980, −1.22718740303448194825563299777, 0, 0, 1.22718740303448194825563299777, 1.24201727822472133475307307980, 2.26029807067196141916295942649, 2.63736641430245155695974597026, 3.43328400577464420303536908336, 3.46953405887859205323107349635, 4.13672942642286505432777441969, 4.91247447846222624710035446709, 5.53019369799135014166152448687, 5.85667632587706229574287450487, 6.93015639305829947630650925503, 7.20076247862321241337333359340, 7.972344528036566289398435306604, 8.078431405365765655048705985109, 8.891119794484000977659631706985, 9.072644867017902083380598720847, 9.895293886851635573468593754445, 10.08531613270476264631197543356

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