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Properties

Label 4-117e2-1.1-c0e2-0-0
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $0.00340946$
Root an. cond. $0.241641$
Motivic weight $0$
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  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-s − 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·7-s − 16-s − 2·19-s + 2·31-s + 2·37-s + 2·49-s − 2·67-s − 2·73-s + 2·97-s − 2·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.00340946\)
Root analytic conductor: \(0.241641\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3101862578\)
\(L(\frac12)\) \(\approx\) \(0.3101862578\)
\(L(1)\) 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)$
bad3 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
5$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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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

−13.80990494819017442544207341531, −13.36001136281506947321561544962, −13.05270677275964737188948525381, −12.72164914596868163355269831842, −11.93499686341665259569750108108, −11.67036785348349858475122944719, −10.65262963256653539214358760704, −10.52450589542211795910627174439, −9.701049372459152198528131483308, −9.460599077616326652932631890719, −8.744582829751551591102352480621, −8.287424301189132676813928040431, −7.37991611206024721121675636931, −6.74227370788945589021286031122, −6.18811586118836933892295665132, −6.02436325627031227717949276801, −4.57945296969333583056588158114, −4.21585912692895745454493259751, −3.10156635655119368499674695018, −2.45453865405386386796231817165, 2.45453865405386386796231817165, 3.10156635655119368499674695018, 4.21585912692895745454493259751, 4.57945296969333583056588158114, 6.02436325627031227717949276801, 6.18811586118836933892295665132, 6.74227370788945589021286031122, 7.37991611206024721121675636931, 8.287424301189132676813928040431, 8.744582829751551591102352480621, 9.460599077616326652932631890719, 9.701049372459152198528131483308, 10.52450589542211795910627174439, 10.65262963256653539214358760704, 11.67036785348349858475122944719, 11.93499686341665259569750108108, 12.72164914596868163355269831842, 13.05270677275964737188948525381, 13.36001136281506947321561544962, 13.80990494819017442544207341531

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