Properties

Label 4-363e2-1.1-c1e2-0-9
Degree $4$
Conductor $131769$
Sign $-1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s − 4-s − 4·6-s + 8·8-s + 9-s − 2·12-s − 7·16-s + 10·17-s − 2·18-s + 16·24-s − 9·25-s − 4·27-s − 18·29-s − 4·31-s − 14·32-s − 20·34-s − 36-s − 6·37-s + 10·41-s − 14·48-s − 10·49-s + 18·50-s + 20·51-s + 8·54-s + 36·58-s + 8·62-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s + 2.82·8-s + 1/3·9-s − 0.577·12-s − 7/4·16-s + 2.42·17-s − 0.471·18-s + 3.26·24-s − 9/5·25-s − 0.769·27-s − 3.34·29-s − 0.718·31-s − 2.47·32-s − 3.42·34-s − 1/6·36-s − 0.986·37-s + 1.56·41-s − 2.02·48-s − 1.42·49-s + 2.54·50-s + 2.80·51-s + 1.08·54-s + 4.72·58-s + 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−9.127942331541421284163370204249, −8.731578058982900025567463262875, −8.264856530445176716222542589828, −7.69035551562908784013462192082, −7.49323810887805876503518242868, −7.43742239849917419501134588307, −6.02373904066748853888838701833, −5.44861874763839175210229504778, −5.21092257719357715807199072115, −3.96049656444482134108637352404, −3.89751875979123562820900427499, −3.26020657517957887232980598752, −1.98218366620434510274683932709, −1.45371564777531463821755560879, 0, 1.45371564777531463821755560879, 1.98218366620434510274683932709, 3.26020657517957887232980598752, 3.89751875979123562820900427499, 3.96049656444482134108637352404, 5.21092257719357715807199072115, 5.44861874763839175210229504778, 6.02373904066748853888838701833, 7.43742239849917419501134588307, 7.49323810887805876503518242868, 7.69035551562908784013462192082, 8.264856530445176716222542589828, 8.731578058982900025567463262875, 9.127942331541421284163370204249

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