Properties

Label 4-111132-1.1-c1e2-0-2
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 $0$

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 + 2·14-s + 5·16-s − 10·25-s + 3·28-s + 12·29-s + 6·32-s + 4·37-s + 16·43-s + 49-s − 20·50-s − 12·53-s + 4·56-s + 24·58-s + 7·64-s − 8·67-s + 8·74-s + 16·79-s + 32·86-s + 2·98-s − 30·100-s − 24·106-s − 24·107-s + 4·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 0.534·14-s + 5/4·16-s − 2·25-s + 0.566·28-s + 2.22·29-s + 1.06·32-s + 0.657·37-s + 2.43·43-s + 1/7·49-s − 2.82·50-s − 1.64·53-s + 0.534·56-s + 3.15·58-s + 7/8·64-s − 0.977·67-s + 0.929·74-s + 1.80·79-s + 3.45·86-s + 0.202·98-s − 3·100-s − 2.33·106-s − 2.32·107-s + 0.383·109-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 111132,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.970518913\)
\(L(\frac12)\) \(\approx\) \(3.970518913\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.393998538973797562285424645673, −9.207333736191082013174831123730, −8.245010328271296459357780366393, −7.80365099617056766338064266882, −7.64010214491199898083268392854, −6.71310127772875365703449938682, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −5.31921059379060528260176389250, −4.74683922157615295493069282209, −4.19102102405251164216473944262, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −2.39101862347891857041372548490, −1.33827299261372059142982522511, 1.33827299261372059142982522511, 2.39101862347891857041372548490, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.19102102405251164216473944262, 4.74683922157615295493069282209, 5.31921059379060528260176389250, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 6.71310127772875365703449938682, 7.64010214491199898083268392854, 7.80365099617056766338064266882, 8.245010328271296459357780366393, 9.207333736191082013174831123730, 9.393998538973797562285424645673

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