Properties

Label 4-640332-1.1-c1e2-0-3
Degree $4$
Conductor $640332$
Sign $1$
Analytic cond. $40.8281$
Root an. cond. $2.52778$
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  + 3-s + 4-s − 2·7-s + 9-s + 12-s − 4·13-s + 16-s − 2·21-s − 6·25-s + 27-s − 2·28-s + 8·31-s + 36-s − 4·37-s − 4·39-s + 8·43-s + 48-s + 3·49-s − 4·52-s − 4·61-s − 2·63-s + 64-s + 24·67-s + 12·73-s − 6·75-s − 16·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.436·21-s − 6/5·25-s + 0.192·27-s − 0.377·28-s + 1.43·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s + 1.21·43-s + 0.144·48-s + 3/7·49-s − 0.554·52-s − 0.512·61-s − 0.251·63-s + 1/8·64-s + 2.93·67-s + 1.40·73-s − 0.692·75-s − 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640332\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(40.8281\)
Root analytic conductor: \(2.52778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 640332,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.146780332\)
\(L(\frac12)\) \(\approx\) \(2.146780332\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 + T )^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.354764684441443853355726283697, −7.921830119955470581232999264379, −7.39315934318606532364206621028, −7.16036475066804637810786253712, −6.50472401562071995733074866821, −6.34557563328331303195978146208, −5.50597076325265524728564634163, −5.33211397644666962563070735994, −4.37951703008965802909913541690, −4.18774926881798203669654007783, −3.37421157013962682769674234557, −2.96523175330119569799799482797, −2.34852765900481584569992467116, −1.88425382150531611231140922836, −0.68984576048428772039915314609, 0.68984576048428772039915314609, 1.88425382150531611231140922836, 2.34852765900481584569992467116, 2.96523175330119569799799482797, 3.37421157013962682769674234557, 4.18774926881798203669654007783, 4.37951703008965802909913541690, 5.33211397644666962563070735994, 5.50597076325265524728564634163, 6.34557563328331303195978146208, 6.50472401562071995733074866821, 7.16036475066804637810786253712, 7.39315934318606532364206621028, 7.921830119955470581232999264379, 8.354764684441443853355726283697

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