Properties

Label 4-2520e2-1.1-c1e2-0-12
Degree $4$
Conductor $6350400$
Sign $1$
Analytic cond. $404.907$
Root an. cond. $4.48578$
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  − 5-s + 5·7-s + 2·11-s + 10·13-s + 4·17-s − 3·19-s − 4·23-s − 8·29-s + 31-s − 5·35-s + 37-s − 12·41-s − 2·43-s + 10·47-s + 18·49-s − 2·55-s + 6·61-s − 10·65-s + 15·67-s + 12·71-s + 3·73-s + 10·77-s − 9·79-s − 12·83-s − 4·85-s + 50·91-s + 3·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.88·7-s + 0.603·11-s + 2.77·13-s + 0.970·17-s − 0.688·19-s − 0.834·23-s − 1.48·29-s + 0.179·31-s − 0.845·35-s + 0.164·37-s − 1.87·41-s − 0.304·43-s + 1.45·47-s + 18/7·49-s − 0.269·55-s + 0.768·61-s − 1.24·65-s + 1.83·67-s + 1.42·71-s + 0.351·73-s + 1.13·77-s − 1.01·79-s − 1.31·83-s − 0.433·85-s + 5.24·91-s + 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6350400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(404.907\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6350400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.139183661\)
\(L(\frac12)\) \(\approx\) \(4.139183661\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.795201482912473612700546105280, −8.707202851721284685348416241559, −8.258743120895068155107739765563, −8.195564172631457662970888699361, −7.57697760880042947896324168853, −7.40810185472724006201442257667, −6.76661022998054266569106021937, −6.38105837694974546616720022035, −5.92132092489142434371408885566, −5.67484172234609576060893685989, −5.15676423351770584615404448627, −4.83216011323019532310344518042, −4.05270387075090431370973319214, −3.86813567945296121499631468035, −3.74410919493692531406211698256, −3.07499212493005001598652338628, −2.08388998352591601357018518138, −1.84499999121165171339269543376, −1.26445358578610905928520409279, −0.77009399449699687942593685180, 0.77009399449699687942593685180, 1.26445358578610905928520409279, 1.84499999121165171339269543376, 2.08388998352591601357018518138, 3.07499212493005001598652338628, 3.74410919493692531406211698256, 3.86813567945296121499631468035, 4.05270387075090431370973319214, 4.83216011323019532310344518042, 5.15676423351770584615404448627, 5.67484172234609576060893685989, 5.92132092489142434371408885566, 6.38105837694974546616720022035, 6.76661022998054266569106021937, 7.40810185472724006201442257667, 7.57697760880042947896324168853, 8.195564172631457662970888699361, 8.258743120895068155107739765563, 8.707202851721284685348416241559, 8.795201482912473612700546105280

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