Properties

Label 4-2592e2-1.1-c1e2-0-34
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s − 4·11-s + 2·13-s − 12·17-s − 8·19-s + 5·25-s − 2·29-s − 4·31-s − 8·35-s − 4·37-s − 2·41-s − 4·43-s − 8·47-s + 7·49-s + 20·53-s + 8·55-s + 4·59-s − 6·61-s − 4·65-s − 4·67-s − 32·71-s − 12·73-s − 16·77-s − 4·79-s − 12·83-s + 24·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s − 2.91·17-s − 1.83·19-s + 25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 0.657·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s + 49-s + 2.74·53-s + 1.07·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 3.79·71-s − 1.40·73-s − 1.82·77-s − 0.450·79-s − 1.31·83-s + 2.60·85-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.731600469424218256517314059154, −8.488638825465717007412615424399, −7.968546495743276285892234566496, −7.55533611077282287995279534489, −7.13486764868327116963586724563, −7.05202789774082241386866214904, −6.21648940658876017122598339493, −6.18837101056719285953327028322, −5.52302718317988151036413742129, −4.86329120212920322314284534981, −4.72316216292546896078046326765, −4.48307130811217325809175245918, −3.93048013535419094775398629864, −3.57776422736899308372712736248, −2.74069323824921438756885998149, −2.33559924612471592659008956244, −1.92578464284909834771248956834, −1.37374890790367982153748551298, 0, 0, 1.37374890790367982153748551298, 1.92578464284909834771248956834, 2.33559924612471592659008956244, 2.74069323824921438756885998149, 3.57776422736899308372712736248, 3.93048013535419094775398629864, 4.48307130811217325809175245918, 4.72316216292546896078046326765, 4.86329120212920322314284534981, 5.52302718317988151036413742129, 6.18837101056719285953327028322, 6.21648940658876017122598339493, 7.05202789774082241386866214904, 7.13486764868327116963586724563, 7.55533611077282287995279534489, 7.968546495743276285892234566496, 8.488638825465717007412615424399, 8.731600469424218256517314059154

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