Properties

Label 4-2e14-1.1-c1e2-0-7
Degree $4$
Conductor $16384$
Sign $1$
Analytic cond. $1.04465$
Root an. cond. $1.01098$
Motivic weight $1$
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  + 2·3-s + 2·5-s + 2·9-s − 2·11-s + 2·13-s + 4·15-s − 4·17-s − 6·19-s + 2·25-s + 6·27-s − 6·29-s − 16·31-s − 4·33-s − 6·37-s + 4·39-s − 10·43-s + 4·45-s + 16·47-s + 10·49-s − 8·51-s + 10·53-s − 4·55-s − 12·57-s + 6·59-s + 18·61-s + 4·65-s + 10·67-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.970·17-s − 1.37·19-s + 2/5·25-s + 1.15·27-s − 1.11·29-s − 2.87·31-s − 0.696·33-s − 0.986·37-s + 0.640·39-s − 1.52·43-s + 0.596·45-s + 2.33·47-s + 10/7·49-s − 1.12·51-s + 1.37·53-s − 0.539·55-s − 1.58·57-s + 0.781·59-s + 2.30·61-s + 0.496·65-s + 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16384\)    =    \(2^{14}\)
Sign: $1$
Analytic conductor: \(1.04465\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16384,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.677614339\)
\(L(\frac12)\) \(\approx\) \(1.677614339\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−13.50457575276406337362419000987, −13.33881577745211123614841678385, −12.55950530458549824198814810685, −12.55925731448152345563002571246, −11.33384213519873814057417509884, −10.98197856559606242788936714678, −10.35542828808534380960988333069, −10.07539591993287036465718180163, −9.066967263146620248958445397648, −8.932410934063420078611538180706, −8.560483771430705131742639013462, −7.80829372269644100130931618851, −6.93434372952847118207164926085, −6.78874395396345968355713133247, −5.52935668894085878039807304958, −5.43719444758836885544966933020, −4.10190629670526837288352010455, −3.66413976188351834094233483200, −2.40795688562533763666136812839, −2.03247494786444624826600944225, 2.03247494786444624826600944225, 2.40795688562533763666136812839, 3.66413976188351834094233483200, 4.10190629670526837288352010455, 5.43719444758836885544966933020, 5.52935668894085878039807304958, 6.78874395396345968355713133247, 6.93434372952847118207164926085, 7.80829372269644100130931618851, 8.560483771430705131742639013462, 8.932410934063420078611538180706, 9.066967263146620248958445397648, 10.07539591993287036465718180163, 10.35542828808534380960988333069, 10.98197856559606242788936714678, 11.33384213519873814057417509884, 12.55925731448152345563002571246, 12.55950530458549824198814810685, 13.33881577745211123614841678385, 13.50457575276406337362419000987

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