Properties

Label 4-1155200-1.1-c1e2-0-7
Degree $4$
Conductor $1155200$
Sign $-1$
Analytic cond. $73.6565$
Root an. cond. $2.92956$
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  + 2·5-s − 2·9-s − 8·13-s − 4·17-s + 3·25-s + 4·29-s + 8·37-s + 12·41-s − 4·45-s + 2·49-s − 16·53-s + 4·61-s − 16·65-s + 12·73-s − 5·81-s − 8·85-s + 28·89-s − 24·97-s + 20·101-s − 4·109-s − 24·113-s + 16·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s − 2.21·13-s − 0.970·17-s + 3/5·25-s + 0.742·29-s + 1.31·37-s + 1.87·41-s − 0.596·45-s + 2/7·49-s − 2.19·53-s + 0.512·61-s − 1.98·65-s + 1.40·73-s − 5/9·81-s − 0.867·85-s + 2.96·89-s − 2.43·97-s + 1.99·101-s − 0.383·109-s − 2.25·113-s + 1.47·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1155200\)    =    \(2^{7} \cdot 5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(73.6565\)
Root analytic conductor: \(2.92956\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1155200,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−7.67065823780629293367938794534, −7.56186383761303726301658586747, −6.75748996467427670181333689729, −6.65071821165806094544770300594, −5.89615972636884024734860870557, −5.81913226127580309415137377006, −4.89054562104167346793520173269, −4.88072322821672453742216928381, −4.37339781441300626459767562156, −3.58012982806994395167456994162, −2.82540491064399701707550154132, −2.40090939476093597229953098979, −2.21004067490187676234569474220, −1.08126566627806166434984305189, 0, 1.08126566627806166434984305189, 2.21004067490187676234569474220, 2.40090939476093597229953098979, 2.82540491064399701707550154132, 3.58012982806994395167456994162, 4.37339781441300626459767562156, 4.88072322821672453742216928381, 4.89054562104167346793520173269, 5.81913226127580309415137377006, 5.89615972636884024734860870557, 6.65071821165806094544770300594, 6.75748996467427670181333689729, 7.56186383761303726301658586747, 7.67065823780629293367938794534

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