Properties

Label 4-627e2-1.1-c1e2-0-0
Degree $4$
Conductor $393129$
Sign $1$
Analytic cond. $25.0662$
Root an. cond. $2.23754$
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  + 2·3-s − 3·4-s − 4·5-s + 3·9-s − 6·12-s − 8·15-s + 5·16-s + 12·20-s + 8·23-s + 2·25-s + 4·27-s + 16·31-s − 9·36-s − 20·37-s − 12·45-s + 24·47-s + 10·48-s − 14·49-s − 12·53-s − 24·59-s + 24·60-s − 3·64-s − 8·67-s + 16·69-s + 4·75-s − 20·80-s + 5·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s − 1.78·5-s + 9-s − 1.73·12-s − 2.06·15-s + 5/4·16-s + 2.68·20-s + 1.66·23-s + 2/5·25-s + 0.769·27-s + 2.87·31-s − 3/2·36-s − 3.28·37-s − 1.78·45-s + 3.50·47-s + 1.44·48-s − 2·49-s − 1.64·53-s − 3.12·59-s + 3.09·60-s − 3/8·64-s − 0.977·67-s + 1.92·69-s + 0.461·75-s − 2.23·80-s + 5/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(393129\)    =    \(3^{2} \cdot 11^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(25.0662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 393129,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9843407694\)
\(L(\frac12)\) \(\approx\) \(0.9843407694\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.653029479594481171374089128382, −8.263513432862467471567669978548, −7.86695452924192516668672252135, −7.52311475462874440391759117602, −7.09492951839214399611242033655, −6.46787589567348650226556085189, −5.77601443425071364494220587649, −4.88480893363690614197292172734, −4.56991321080597356703354311172, −4.43128249952073642221709631783, −3.48982533128044565942915605098, −3.44365518362789223021999993571, −2.81805056436224670768266257292, −1.60994939898295228077229523210, −0.54470855901702999376946866933, 0.54470855901702999376946866933, 1.60994939898295228077229523210, 2.81805056436224670768266257292, 3.44365518362789223021999993571, 3.48982533128044565942915605098, 4.43128249952073642221709631783, 4.56991321080597356703354311172, 4.88480893363690614197292172734, 5.77601443425071364494220587649, 6.46787589567348650226556085189, 7.09492951839214399611242033655, 7.52311475462874440391759117602, 7.86695452924192516668672252135, 8.263513432862467471567669978548, 8.653029479594481171374089128382

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