Properties

Label 4-111132-1.1-c1e2-0-4
Degree $4$
Conductor $111132$
Sign $-1$
Analytic cond. $7.08587$
Root an. cond. $1.63154$
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 + 3·4-s − 7-s − 4·8-s + 8·11-s + 2·14-s + 5·16-s − 16·22-s − 16·23-s − 6·25-s − 3·28-s + 4·29-s − 6·32-s − 20·37-s − 8·43-s + 24·44-s + 32·46-s + 49-s + 12·50-s − 12·53-s + 4·56-s − 8·58-s + 7·64-s + 8·67-s − 16·71-s + 40·74-s − 8·77-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 2.41·11-s + 0.534·14-s + 5/4·16-s − 3.41·22-s − 3.33·23-s − 6/5·25-s − 0.566·28-s + 0.742·29-s − 1.06·32-s − 3.28·37-s − 1.21·43-s + 3.61·44-s + 4.71·46-s + 1/7·49-s + 1.69·50-s − 1.64·53-s + 0.534·56-s − 1.05·58-s + 7/8·64-s + 0.977·67-s − 1.89·71-s + 4.64·74-s − 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(111132\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(7.08587\)
Root analytic conductor: \(1.63154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{111132} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 111132,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.164662289745378790951914691450, −8.882543700432626086795887126363, −8.337652051623380123011624806396, −7.954464142207805330736993023418, −7.36399840290663473896741338506, −6.58813712157544804919343144385, −6.54855688605176886120203881032, −6.01899647743800981343957937964, −5.32455191890654195761811472253, −4.16943092436514571048096348777, −3.83863171110134241641513411613, −3.17941538941685381741709304651, −1.79365373540820646322667624458, −1.71238689353652455033144003099, 0, 1.71238689353652455033144003099, 1.79365373540820646322667624458, 3.17941538941685381741709304651, 3.83863171110134241641513411613, 4.16943092436514571048096348777, 5.32455191890654195761811472253, 6.01899647743800981343957937964, 6.54855688605176886120203881032, 6.58813712157544804919343144385, 7.36399840290663473896741338506, 7.954464142207805330736993023418, 8.337652051623380123011624806396, 8.882543700432626086795887126363, 9.164662289745378790951914691450

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