Properties

Label 4-1095200-1.1-c1e2-0-12
Degree $4$
Conductor $1095200$
Sign $-1$
Analytic cond. $69.8309$
Root an. cond. $2.89075$
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-s + 4-s + 2·5-s − 8-s − 2·9-s − 2·10-s + 16-s + 6·17-s + 2·18-s + 2·20-s + 3·25-s − 6·29-s − 32-s − 6·34-s − 2·36-s − 2·37-s − 2·40-s + 6·41-s − 4·45-s − 13·49-s − 3·50-s + 26·53-s + 6·58-s − 30·61-s + 64-s + 6·68-s + 2·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 1/4·16-s + 1.45·17-s + 0.471·18-s + 0.447·20-s + 3/5·25-s − 1.11·29-s − 0.176·32-s − 1.02·34-s − 1/3·36-s − 0.328·37-s − 0.316·40-s + 0.937·41-s − 0.596·45-s − 1.85·49-s − 0.424·50-s + 3.57·53-s + 0.787·58-s − 3.84·61-s + 1/8·64-s + 0.727·68-s + 0.235·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1095200\)    =    \(2^{5} \cdot 5^{2} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(69.8309\)
Root analytic conductor: \(2.89075\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1095200,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−7.947480595907964918208604648618, −7.49831009051549166645447960646, −6.89931657836489782864864420449, −6.76654201641600851167649939545, −5.77272193873314481326296907642, −5.77162846845092713605544691251, −5.55177590831197844606272911275, −4.75848365306684889407447061330, −4.12586421650220620613319696777, −3.52332963371328713101003529737, −2.82505147340032154850850367219, −2.62513400004584128215759559012, −1.63805431218632083963465992264, −1.26592217580020237546694300482, 0, 1.26592217580020237546694300482, 1.63805431218632083963465992264, 2.62513400004584128215759559012, 2.82505147340032154850850367219, 3.52332963371328713101003529737, 4.12586421650220620613319696777, 4.75848365306684889407447061330, 5.55177590831197844606272911275, 5.77162846845092713605544691251, 5.77272193873314481326296907642, 6.76654201641600851167649939545, 6.89931657836489782864864420449, 7.49831009051549166645447960646, 7.947480595907964918208604648618

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