Properties

Label 4-3240e2-1.1-c1e2-0-33
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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 − 2·7-s − 4·11-s + 2·13-s + 10·17-s − 10·19-s − 23-s + 2·29-s − 7·31-s + 2·35-s − 12·37-s − 4·43-s − 4·47-s + 7·49-s + 18·53-s + 4·55-s − 14·59-s + 11·61-s − 2·65-s − 14·67-s − 24·73-s + 8·77-s + 3·79-s + 83-s − 10·85-s − 4·91-s + 10·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s + 2.42·17-s − 2.29·19-s − 0.208·23-s + 0.371·29-s − 1.25·31-s + 0.338·35-s − 1.97·37-s − 0.609·43-s − 0.583·47-s + 49-s + 2.47·53-s + 0.539·55-s − 1.82·59-s + 1.40·61-s − 0.248·65-s − 1.71·67-s − 2.80·73-s + 0.911·77-s + 0.337·79-s + 0.109·83-s − 1.08·85-s − 0.419·91-s + 1.02·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 16 T + 159 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−8.426214136173549626823674681205, −8.221461241786966854261183429566, −7.74102598348565152369449584586, −7.34717601267661802066936726957, −6.93475415823980426743478386688, −6.79260929886829257536748545985, −6.04424274637735167775994557677, −5.84089177059906593529320536819, −5.40564780647186887300642396942, −5.24571667797557006126350836153, −4.47582306442265481942921054047, −4.13132947735132223594679723017, −3.61993608192267871246527396146, −3.43473305800243617601064676474, −2.75138002872948096609710356761, −2.52708640949624681869371476409, −1.66291354796525308838072874263, −1.26675937112357791425143162767, 0, 0, 1.26675937112357791425143162767, 1.66291354796525308838072874263, 2.52708640949624681869371476409, 2.75138002872948096609710356761, 3.43473305800243617601064676474, 3.61993608192267871246527396146, 4.13132947735132223594679723017, 4.47582306442265481942921054047, 5.24571667797557006126350836153, 5.40564780647186887300642396942, 5.84089177059906593529320536819, 6.04424274637735167775994557677, 6.79260929886829257536748545985, 6.93475415823980426743478386688, 7.34717601267661802066936726957, 7.74102598348565152369449584586, 8.221461241786966854261183429566, 8.426214136173549626823674681205

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