Properties

Label 4-1440e2-1.1-c1e2-0-53
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·13-s − 4·17-s + 3·25-s − 12·29-s − 20·37-s − 4·41-s − 6·49-s − 12·53-s − 4·61-s + 8·65-s − 12·73-s + 8·85-s − 20·89-s + 4·97-s + 4·101-s − 36·109-s − 4·113-s + 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.10·13-s − 0.970·17-s + 3/5·25-s − 2.22·29-s − 3.28·37-s − 0.624·41-s − 6/7·49-s − 1.64·53-s − 0.512·61-s + 0.992·65-s − 1.40·73-s + 0.867·85-s − 2.11·89-s + 0.406·97-s + 0.398·101-s − 3.44·109-s − 0.376·113-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−9.206883926168802619577668862186, −8.998316881458210800743197178988, −8.459899785669851719256164874973, −8.211255681173920907997826197619, −7.59458585749141650322842663588, −7.35586860808036520391291120827, −6.95515378857305272642017619496, −6.66330364534288339791213305386, −6.07074588456278407762999436348, −5.44504565512475843427347052581, −5.09877413997665161186609046661, −4.77931011777860238276576962090, −4.06739465071637553476845919884, −3.86485250752345530176479745830, −3.14880370233397003691309117407, −2.84633864821837641665232197750, −1.82540296496244342363513955589, −1.69086024587047792801753188621, 0, 0, 1.69086024587047792801753188621, 1.82540296496244342363513955589, 2.84633864821837641665232197750, 3.14880370233397003691309117407, 3.86485250752345530176479745830, 4.06739465071637553476845919884, 4.77931011777860238276576962090, 5.09877413997665161186609046661, 5.44504565512475843427347052581, 6.07074588456278407762999436348, 6.66330364534288339791213305386, 6.95515378857305272642017619496, 7.35586860808036520391291120827, 7.59458585749141650322842663588, 8.211255681173920907997826197619, 8.459899785669851719256164874973, 8.998316881458210800743197178988, 9.206883926168802619577668862186

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