Properties

Label 4-6080e2-1.1-c1e2-0-11
Degree $4$
Conductor $36966400$
Sign $1$
Analytic cond. $2357.00$
Root an. cond. $6.96771$
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  − 2·3-s − 2·5-s + 4·7-s + 2·13-s + 4·15-s − 2·19-s − 8·21-s + 3·25-s + 2·27-s + 4·31-s − 8·35-s − 10·37-s − 4·39-s − 12·41-s − 4·43-s + 12·47-s − 2·49-s + 18·53-s + 4·57-s − 4·61-s − 4·65-s − 10·67-s − 12·71-s − 8·73-s − 6·75-s − 8·79-s − 81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1.51·7-s + 0.554·13-s + 1.03·15-s − 0.458·19-s − 1.74·21-s + 3/5·25-s + 0.384·27-s + 0.718·31-s − 1.35·35-s − 1.64·37-s − 0.640·39-s − 1.87·41-s − 0.609·43-s + 1.75·47-s − 2/7·49-s + 2.47·53-s + 0.529·57-s − 0.512·61-s − 0.496·65-s − 1.22·67-s − 1.42·71-s − 0.936·73-s − 0.692·75-s − 0.900·79-s − 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(36966400\)    =    \(2^{12} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2357.00\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 36966400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 154 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.84501182151761096713105601714, −7.41716731211517361515842089102, −7.20883961411822483944228841817, −6.99564804252974805466901702016, −6.33379901165874026266606079897, −6.06751009669807373080870752007, −5.79916595798146563185807810262, −5.24869759704150111834635173248, −5.06957600138719121341195212249, −4.75152315655695434093315638072, −4.29812303470586571476454127915, −3.97894299381630711573307858105, −3.53673484784191877867127482751, −3.06566300237167678146767588062, −2.53590389288630047714337459904, −1.96543987024225144570243801060, −1.35235105511538362174068784132, −1.14736107804908747217697750563, 0, 0, 1.14736107804908747217697750563, 1.35235105511538362174068784132, 1.96543987024225144570243801060, 2.53590389288630047714337459904, 3.06566300237167678146767588062, 3.53673484784191877867127482751, 3.97894299381630711573307858105, 4.29812303470586571476454127915, 4.75152315655695434093315638072, 5.06957600138719121341195212249, 5.24869759704150111834635173248, 5.79916595798146563185807810262, 6.06751009669807373080870752007, 6.33379901165874026266606079897, 6.99564804252974805466901702016, 7.20883961411822483944228841817, 7.41716731211517361515842089102, 7.84501182151761096713105601714

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