Properties

Label 4-532e2-1.1-c1e2-0-6
Degree $4$
Conductor $283024$
Sign $1$
Analytic cond. $18.0458$
Root an. cond. $2.06107$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 6·5-s + 2·7-s − 3·11-s − 2·13-s − 6·15-s + 3·17-s + 2·19-s − 2·21-s + 17·25-s − 2·27-s + 3·29-s + 31-s + 3·33-s + 12·35-s + 10·37-s + 2·39-s − 3·41-s + 4·43-s + 12·47-s + 3·49-s − 3·51-s − 3·53-s − 18·55-s − 2·57-s + 6·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.68·5-s + 0.755·7-s − 0.904·11-s − 0.554·13-s − 1.54·15-s + 0.727·17-s + 0.458·19-s − 0.436·21-s + 17/5·25-s − 0.384·27-s + 0.557·29-s + 0.179·31-s + 0.522·33-s + 2.02·35-s + 1.64·37-s + 0.320·39-s − 0.468·41-s + 0.609·43-s + 1.75·47-s + 3/7·49-s − 0.420·51-s − 0.412·53-s − 2.42·55-s − 0.264·57-s + 0.781·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(283024\)    =    \(2^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(18.0458\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 283024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.637119875\)
\(L(\frac12)\) \(\approx\) \(2.637119875\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 5 T + 93 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 19 T + 189 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 15 T + 175 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 166 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−10.65956759752428658695631982286, −10.64758075775048259794936033763, −10.27252764473795337469734156708, −9.671369471696962735996023165166, −9.427178393871595796018542308753, −9.209891857393727686192046214456, −8.291756036876013472680338059872, −8.025902644879022791631402380467, −7.38138449244145568345351900091, −6.93206250461869467950400949115, −6.15569878512594788467150208518, −5.98232778663916501699604567658, −5.37300311549347629519308899045, −5.34489458742315857556074066981, −4.70845617294110711937170327661, −3.96682945915399810954508145916, −2.64284162634064698467222211686, −2.64136758039257950310911705616, −1.79012099402188512581923391590, −1.07997414835013747402324904307, 1.07997414835013747402324904307, 1.79012099402188512581923391590, 2.64136758039257950310911705616, 2.64284162634064698467222211686, 3.96682945915399810954508145916, 4.70845617294110711937170327661, 5.34489458742315857556074066981, 5.37300311549347629519308899045, 5.98232778663916501699604567658, 6.15569878512594788467150208518, 6.93206250461869467950400949115, 7.38138449244145568345351900091, 8.025902644879022791631402380467, 8.291756036876013472680338059872, 9.209891857393727686192046214456, 9.427178393871595796018542308753, 9.671369471696962735996023165166, 10.27252764473795337469734156708, 10.64758075775048259794936033763, 10.65956759752428658695631982286

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