Properties

Label 4-700e2-1.1-c1e2-0-6
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 6·11-s − 4·19-s − 18·29-s + 16·31-s − 12·41-s − 49-s − 20·61-s + 30·71-s + 14·79-s − 5·81-s + 24·89-s + 12·99-s + 36·101-s + 14·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·171-s + ⋯
L(s)  = 1  + 2/3·9-s + 1.80·11-s − 0.917·19-s − 3.34·29-s + 2.87·31-s − 1.87·41-s − 1/7·49-s − 2.56·61-s + 3.56·71-s + 1.57·79-s − 5/9·81-s + 2.54·89-s + 1.20·99-s + 3.58·101-s + 1.34·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.611·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.137763132\)
\(L(\frac12)\) \(\approx\) \(2.137763132\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.65137952057750505305899590694, −10.22878760541338227363484868375, −9.701215662760807156810281907471, −9.250210507348614390827957263560, −9.244706208696706381131837475132, −8.437478049592419141278881755028, −8.179829966329081059073020805028, −7.57488352665804640850623102093, −7.12682876507005590906700878798, −6.64545750706687171589924825427, −6.24198449306097599567071623280, −6.00621160863797576205116566336, −5.10995238471114534684013773628, −4.65311937294734594598776842469, −4.23410994602927263536481362554, −3.46289100504918433249554152372, −3.44655202647465720328780809249, −2.05982412550434002318841970182, −1.83553176259099801294864750371, −0.805558116170743570308512986130, 0.805558116170743570308512986130, 1.83553176259099801294864750371, 2.05982412550434002318841970182, 3.44655202647465720328780809249, 3.46289100504918433249554152372, 4.23410994602927263536481362554, 4.65311937294734594598776842469, 5.10995238471114534684013773628, 6.00621160863797576205116566336, 6.24198449306097599567071623280, 6.64545750706687171589924825427, 7.12682876507005590906700878798, 7.57488352665804640850623102093, 8.179829966329081059073020805028, 8.437478049592419141278881755028, 9.244706208696706381131837475132, 9.250210507348614390827957263560, 9.701215662760807156810281907471, 10.22878760541338227363484868375, 10.65137952057750505305899590694

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