Properties

Label 4-2592e2-1.1-c1e2-0-14
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  + 2·5-s + 7-s − 2·11-s − 13-s − 12·17-s + 10·19-s + 6·23-s + 5·25-s + 8·29-s + 8·31-s + 2·35-s − 10·37-s + 8·41-s − 4·43-s − 10·47-s + 7·49-s − 8·53-s − 4·55-s + 14·59-s − 3·61-s − 2·65-s − 13·67-s + 8·71-s + 18·73-s − 2·77-s + 11·79-s − 12·83-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 0.603·11-s − 0.277·13-s − 2.91·17-s + 2.29·19-s + 1.25·23-s + 25-s + 1.48·29-s + 1.43·31-s + 0.338·35-s − 1.64·37-s + 1.24·41-s − 0.609·43-s − 1.45·47-s + 49-s − 1.09·53-s − 0.539·55-s + 1.82·59-s − 0.384·61-s − 0.248·65-s − 1.58·67-s + 0.949·71-s + 2.10·73-s − 0.227·77-s + 1.23·79-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2592} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.015623792\)
\(L(\frac12)\) \(\approx\) \(3.015623792\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−9.213002221579420823555174865191, −8.749620002457753040744826696389, −8.219724971534786197642094283478, −8.165086835023029320944794891760, −7.47353419591047816088231299053, −7.05160048666413324128826862176, −6.74997269872407764580730962900, −6.48146365822929023586585994050, −6.13195104448681677519464048472, −5.27194394988949148319469265090, −5.08740616703089938654967419247, −5.03824587906532828814171391247, −4.42350631041778092247109224710, −3.96671119725420536024512769369, −3.14992470577573226353437722162, −2.81009910544668452973445177859, −2.50717646661653915329231785718, −1.88150168958661787213519950379, −1.26214441446488709386656294056, −0.60258635956818946545576541919, 0.60258635956818946545576541919, 1.26214441446488709386656294056, 1.88150168958661787213519950379, 2.50717646661653915329231785718, 2.81009910544668452973445177859, 3.14992470577573226353437722162, 3.96671119725420536024512769369, 4.42350631041778092247109224710, 5.03824587906532828814171391247, 5.08740616703089938654967419247, 5.27194394988949148319469265090, 6.13195104448681677519464048472, 6.48146365822929023586585994050, 6.74997269872407764580730962900, 7.05160048666413324128826862176, 7.47353419591047816088231299053, 8.165086835023029320944794891760, 8.219724971534786197642094283478, 8.749620002457753040744826696389, 9.213002221579420823555174865191

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