Properties

Label 4-1095200-1.1-c1e2-0-10
Degree $4$
Conductor $1095200$
Sign $-1$
Analytic cond. $69.8309$
Root an. cond. $2.89075$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s + 4·9-s − 10-s − 3·13-s + 16-s − 4·18-s + 20-s − 4·25-s + 3·26-s − 32-s + 4·36-s − 8·37-s − 40-s − 9·41-s + 4·45-s + 5·49-s + 4·50-s − 3·52-s − 7·53-s + 5·61-s + 64-s − 3·65-s − 4·72-s + 8·74-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s + 4/3·9-s − 0.316·10-s − 0.832·13-s + 1/4·16-s − 0.942·18-s + 0.223·20-s − 4/5·25-s + 0.588·26-s − 0.176·32-s + 2/3·36-s − 1.31·37-s − 0.158·40-s − 1.40·41-s + 0.596·45-s + 5/7·49-s + 0.565·50-s − 0.416·52-s − 0.961·53-s + 0.640·61-s + 1/8·64-s − 0.372·65-s − 0.471·72-s + 0.929·74-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: yes
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$C_2$ \( 1 - T + p T^{2} \)
37$C_2$ \( 1 + 8 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.72046659580019593313543126522, −7.44233739171997135907955976169, −7.10477651159503319675021511502, −6.59418048139677526840874061479, −6.32835214985814926083945823243, −5.58984240437669804435224876506, −5.22759714613272812709239763260, −4.73085134156516446637164026689, −4.15671404059170282010540179471, −3.63536360159906566313247530381, −3.03424934942362814097272611877, −2.24854289456122829340209902841, −1.82870692585144507262393023918, −1.20507569496527294566002758393, 0, 1.20507569496527294566002758393, 1.82870692585144507262393023918, 2.24854289456122829340209902841, 3.03424934942362814097272611877, 3.63536360159906566313247530381, 4.15671404059170282010540179471, 4.73085134156516446637164026689, 5.22759714613272812709239763260, 5.58984240437669804435224876506, 6.32835214985814926083945823243, 6.59418048139677526840874061479, 7.10477651159503319675021511502, 7.44233739171997135907955976169, 7.72046659580019593313543126522

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