Properties

Label 4-557568-1.1-c1e2-0-72
Degree $4$
Conductor $557568$
Sign $-1$
Analytic cond. $35.5510$
Root an. cond. $2.44181$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 4·11-s − 4·13-s − 6·25-s − 4·27-s + 12·29-s − 8·33-s + 8·39-s − 14·49-s + 8·59-s − 4·61-s − 8·67-s + 12·75-s − 16·79-s + 5·81-s − 24·87-s − 12·89-s + 4·97-s + 12·99-s − 36·101-s − 4·109-s + 36·113-s − 12·117-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 1.20·11-s − 1.10·13-s − 6/5·25-s − 0.769·27-s + 2.22·29-s − 1.39·33-s + 1.28·39-s − 2·49-s + 1.04·59-s − 0.512·61-s − 0.977·67-s + 1.38·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s − 1.27·89-s + 0.406·97-s + 1.20·99-s − 3.58·101-s − 0.383·109-s + 3.38·113-s − 1.10·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(557568\)    =    \(2^{9} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(35.5510\)
Root analytic conductor: \(2.44181\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 557568,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.098990694093691505068092710868, −7.82048755073253042108636697295, −7.04619174160680993198292873454, −6.86309261643108304364881933941, −6.42897107072744719896577758454, −5.94195259388428776855182860466, −5.49024592405004234969144901480, −4.94454899743565935082637503624, −4.33990220358944048865857917052, −4.25303028692796488061253338187, −3.32045065868348143323558177944, −2.71136660887374010040841871572, −1.80957061896546408108660751255, −1.14203459290393266350032133514, 0, 1.14203459290393266350032133514, 1.80957061896546408108660751255, 2.71136660887374010040841871572, 3.32045065868348143323558177944, 4.25303028692796488061253338187, 4.33990220358944048865857917052, 4.94454899743565935082637503624, 5.49024592405004234969144901480, 5.94195259388428776855182860466, 6.42897107072744719896577758454, 6.86309261643108304364881933941, 7.04619174160680993198292873454, 7.82048755073253042108636697295, 8.098990694093691505068092710868

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