Properties

Label 4-672e2-1.1-c1e2-0-87
Degree $4$
Conductor $451584$
Sign $1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 9-s − 12·13-s − 4·17-s + 2·25-s − 4·29-s − 20·37-s − 4·41-s − 4·45-s + 49-s − 20·53-s + 20·61-s + 48·65-s + 4·73-s + 81-s + 16·85-s + 12·89-s + 4·97-s − 20·101-s − 4·109-s − 28·113-s − 12·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.78·5-s + 1/3·9-s − 3.32·13-s − 0.970·17-s + 2/5·25-s − 0.742·29-s − 3.28·37-s − 0.624·41-s − 0.596·45-s + 1/7·49-s − 2.74·53-s + 2.56·61-s + 5.95·65-s + 0.468·73-s + 1/9·81-s + 1.73·85-s + 1.27·89-s + 0.406·97-s − 1.99·101-s − 0.383·109-s − 2.63·113-s − 1.10·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{451584} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.034900753827151172045193415181, −7.70276610727881458590413914623, −7.20214522889553119059062687435, −6.78044077445841975719051191645, −6.77473142578757723529467681346, −5.36857771747617040863256466681, −5.33934975205348273864801508199, −4.52969287471330622027217048773, −4.44304823928958265064368122586, −3.61417461907435968748262507791, −3.27099070744047974080065927216, −2.32898343675991772184414351496, −1.90612310330978239121510440566, 0, 0, 1.90612310330978239121510440566, 2.32898343675991772184414351496, 3.27099070744047974080065927216, 3.61417461907435968748262507791, 4.44304823928958265064368122586, 4.52969287471330622027217048773, 5.33934975205348273864801508199, 5.36857771747617040863256466681, 6.77473142578757723529467681346, 6.78044077445841975719051191645, 7.20214522889553119059062687435, 7.70276610727881458590413914623, 8.034900753827151172045193415181

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