Properties

Label 4-537e2-1.1-c1e2-0-2
Degree $4$
Conductor $288369$
Sign $1$
Analytic cond. $18.3866$
Root an. cond. $2.07074$
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  − 8·7-s − 3·9-s − 2·13-s − 4·16-s − 6·19-s − 25-s − 16·31-s + 4·37-s − 22·43-s + 34·49-s + 28·61-s + 24·63-s − 18·67-s + 20·73-s + 20·79-s + 9·81-s + 16·91-s − 28·97-s − 12·103-s − 28·109-s + 32·112-s + 6·117-s − 6·121-s + 127-s + 131-s + 48·133-s + 137-s + ⋯
L(s)  = 1  − 3.02·7-s − 9-s − 0.554·13-s − 16-s − 1.37·19-s − 1/5·25-s − 2.87·31-s + 0.657·37-s − 3.35·43-s + 34/7·49-s + 3.58·61-s + 3.02·63-s − 2.19·67-s + 2.34·73-s + 2.25·79-s + 81-s + 1.67·91-s − 2.84·97-s − 1.18·103-s − 2.68·109-s + 3.02·112-s + 0.554·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(288369\)    =    \(3^{2} \cdot 179^{2}\)
Sign: $1$
Analytic conductor: \(18.3866\)
Root analytic conductor: \(2.07074\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 288369,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_2$ \( 1 + p T^{2} \)
179$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
89$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.410709202677624902364232584133, −8.096748629455889385157419531865, −7.16979390191822950971725375612, −6.69315229163834238910355551443, −6.68403281767003921511335972766, −6.18510863780788689281989494770, −5.39698209592502247550241628253, −5.26927125289354544444142895540, −4.14390277306985977394407953592, −3.65455224933898053965548391576, −3.30834137302291977565583010941, −2.57742433461389850447314956434, −2.11337535992451098529704572807, 0, 0, 2.11337535992451098529704572807, 2.57742433461389850447314956434, 3.30834137302291977565583010941, 3.65455224933898053965548391576, 4.14390277306985977394407953592, 5.26927125289354544444142895540, 5.39698209592502247550241628253, 6.18510863780788689281989494770, 6.68403281767003921511335972766, 6.69315229163834238910355551443, 7.16979390191822950971725375612, 8.096748629455889385157419531865, 8.410709202677624902364232584133

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