Properties

Label 4-798e2-1.1-c1e2-0-89
Degree $4$
Conductor $636804$
Sign $-1$
Analytic cond. $40.6031$
Root an. cond. $2.52429$
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  + 2·2-s + 3·4-s − 4·7-s + 4·8-s + 9-s − 8·14-s + 5·16-s + 2·18-s − 12·23-s − 10·25-s − 12·28-s + 12·29-s + 6·32-s + 3·36-s − 8·37-s − 8·43-s − 24·46-s + 9·49-s − 20·50-s + 12·53-s − 16·56-s + 24·58-s − 4·63-s + 7·64-s + 16·67-s + 4·72-s − 16·74-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s + 1/3·9-s − 2.13·14-s + 5/4·16-s + 0.471·18-s − 2.50·23-s − 2·25-s − 2.26·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s − 1.31·37-s − 1.21·43-s − 3.53·46-s + 9/7·49-s − 2.82·50-s + 1.64·53-s − 2.13·56-s + 3.15·58-s − 0.503·63-s + 7/8·64-s + 1.95·67-s + 0.471·72-s − 1.85·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(636804\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(40.6031\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{636804} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 636804,\ (\ :1/2, 1/2),\ -1)\)

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

−7.973335490421157373200340268575, −7.70397779901611905763392602462, −6.84732233829542762166868447522, −6.67685453158503036830492818980, −6.39461130745147042664210509886, −5.65473838318916067878138758246, −5.61719283222003359694793802526, −4.85679242074249456985197372563, −4.12049989296715617337624484159, −3.89674503164637365373404147869, −3.52845307997211445678021341580, −2.72251484544452322835217288408, −2.32322030951675246311285954744, −1.51207835751420731285626782871, 0, 1.51207835751420731285626782871, 2.32322030951675246311285954744, 2.72251484544452322835217288408, 3.52845307997211445678021341580, 3.89674503164637365373404147869, 4.12049989296715617337624484159, 4.85679242074249456985197372563, 5.61719283222003359694793802526, 5.65473838318916067878138758246, 6.39461130745147042664210509886, 6.67685453158503036830492818980, 6.84732233829542762166868447522, 7.70397779901611905763392602462, 7.973335490421157373200340268575

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