Properties

Label 4-798e2-1.1-c1e2-0-72
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 2·7-s + 4·8-s + 9-s + 6·12-s + 4·14-s + 5·16-s + 2·18-s + 2·19-s + 4·21-s + 8·24-s − 10·25-s − 4·27-s + 6·28-s + 12·29-s + 6·32-s + 3·36-s + 4·38-s − 12·41-s + 8·42-s + 16·43-s + 10·48-s + 3·49-s − 20·50-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 1/3·9-s + 1.73·12-s + 1.06·14-s + 5/4·16-s + 0.471·18-s + 0.458·19-s + 0.872·21-s + 1.63·24-s − 2·25-s − 0.769·27-s + 1.13·28-s + 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.648·38-s − 1.87·41-s + 1.23·42-s + 2.43·43-s + 1.44·48-s + 3/7·49-s − 2.82·50-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: \(0\)
Selberg data: \((4,\ 636804,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.377584104\)
\(L(\frac12)\) \(\approx\) \(8.377584104\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_2$ \( 1 - 2 T + p T^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−8.196197307405668368216926690290, −7.80365099617056766338064266882, −7.72200516209190044787449977815, −6.77298005536370046551425088692, −6.71310127772875365703449938682, −5.97907088754740589421145166260, −5.31921059379060528260176389250, −5.31155422432102619146983664079, −4.35906178911561015019581069777, −4.19102102405251164216473944262, −3.55246093072316681260396868353, −3.09738642858536778661614069445, −2.39101862347891857041372548490, −2.11458272020859388026266946767, −1.19460132813703026742735308794, 1.19460132813703026742735308794, 2.11458272020859388026266946767, 2.39101862347891857041372548490, 3.09738642858536778661614069445, 3.55246093072316681260396868353, 4.19102102405251164216473944262, 4.35906178911561015019581069777, 5.31155422432102619146983664079, 5.31921059379060528260176389250, 5.97907088754740589421145166260, 6.71310127772875365703449938682, 6.77298005536370046551425088692, 7.72200516209190044787449977815, 7.80365099617056766338064266882, 8.196197307405668368216926690290

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