Properties

Label 4-2300e2-1.1-c1e2-0-2
Degree $4$
Conductor $5290000$
Sign $1$
Analytic cond. $337.294$
Root an. cond. $4.28550$
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  + 2·9-s − 6·11-s + 14·19-s − 6·29-s + 4·31-s − 18·41-s + 13·49-s + 16·61-s + 24·71-s + 2·79-s − 5·81-s − 24·89-s − 12·99-s − 36·101-s + 20·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 28·171-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.80·11-s + 3.21·19-s − 1.11·29-s + 0.718·31-s − 2.81·41-s + 13/7·49-s + 2.04·61-s + 2.84·71-s + 0.225·79-s − 5/9·81-s − 2.54·89-s − 1.20·99-s − 3.58·101-s + 1.91·109-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 + 1/13·169-s + 2.14·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5290000\)    =    \(2^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(337.294\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.196874248\)
\(L(\frac12)\) \(\approx\) \(2.196874248\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 157 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.590937286736096151187154469119, −8.680177336998607085210956755922, −8.372024652520131201398334723211, −7.924309652060495646754294412391, −7.81658027910585207906520753052, −7.07812042458330418845295150192, −7.04046960550593111726012278933, −6.80720631877851710002389233872, −5.80204595966331981999937397132, −5.54535929699843176567302338440, −5.28786086242878564729896802403, −5.03502906029712633535952604299, −4.45428764512711192257630814849, −3.75258570683351456117377164429, −3.51744978628669179456907996735, −2.90910288751077892164335207015, −2.57242611537364738274434436901, −1.87789431933432821500464787496, −1.23572236365986231043252581050, −0.54864507796676767195401108838, 0.54864507796676767195401108838, 1.23572236365986231043252581050, 1.87789431933432821500464787496, 2.57242611537364738274434436901, 2.90910288751077892164335207015, 3.51744978628669179456907996735, 3.75258570683351456117377164429, 4.45428764512711192257630814849, 5.03502906029712633535952604299, 5.28786086242878564729896802403, 5.54535929699843176567302338440, 5.80204595966331981999937397132, 6.80720631877851710002389233872, 7.04046960550593111726012278933, 7.07812042458330418845295150192, 7.81658027910585207906520753052, 7.924309652060495646754294412391, 8.372024652520131201398334723211, 8.680177336998607085210956755922, 9.590937286736096151187154469119

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