Properties

Label 4-1368e2-1.1-c1e2-0-25
Degree $4$
Conductor $1871424$
Sign $1$
Analytic cond. $119.323$
Root an. cond. $3.30507$
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  − 5-s + 7-s − 7·11-s − 2·13-s − 17-s + 2·19-s − 10·23-s − 5·25-s − 4·31-s − 35-s − 16·37-s + 2·41-s + 43-s + 47-s − 9·49-s + 8·53-s + 7·55-s − 24·59-s + 61-s + 2·65-s − 4·67-s − 5·73-s − 7·77-s − 2·79-s − 8·83-s + 85-s + 12·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 2.11·11-s − 0.554·13-s − 0.242·17-s + 0.458·19-s − 2.08·23-s − 25-s − 0.718·31-s − 0.169·35-s − 2.63·37-s + 0.312·41-s + 0.152·43-s + 0.145·47-s − 9/7·49-s + 1.09·53-s + 0.943·55-s − 3.12·59-s + 0.128·61-s + 0.248·65-s − 0.488·67-s − 0.585·73-s − 0.797·77-s − 0.225·79-s − 0.878·83-s + 0.108·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(119.323\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1368} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1871424,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T + 56 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - T + 84 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 5 T - 56 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−9.202662513579733759431425021264, −9.149236607541663322133941012394, −8.411515746775814653294218273641, −8.084711086656233871244845227047, −7.66190770990152056405132713219, −7.65332501703342613470881963732, −7.12119993922417367778747557959, −6.50324011812014693129658106225, −5.98630505389494854275615366728, −5.58047950761476600184648104997, −5.05819907741072692268536925856, −4.98739524312174120048643645785, −4.02883090227823545932008734574, −4.00395169995047673255636676435, −3.06940715612023451468291156546, −2.80203519824726207782574024181, −1.94093804883590667586111335914, −1.72180829274827917643095105983, 0, 0, 1.72180829274827917643095105983, 1.94093804883590667586111335914, 2.80203519824726207782574024181, 3.06940715612023451468291156546, 4.00395169995047673255636676435, 4.02883090227823545932008734574, 4.98739524312174120048643645785, 5.05819907741072692268536925856, 5.58047950761476600184648104997, 5.98630505389494854275615366728, 6.50324011812014693129658106225, 7.12119993922417367778747557959, 7.65332501703342613470881963732, 7.66190770990152056405132713219, 8.084711086656233871244845227047, 8.411515746775814653294218273641, 9.149236607541663322133941012394, 9.202662513579733759431425021264

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