Properties

Label 4-1815e2-1.1-c1e2-0-1
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
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  + 3·4-s − 2·5-s − 9-s + 5·16-s − 16·19-s − 6·20-s − 25-s − 8·29-s − 16·31-s − 3·36-s + 24·41-s + 2·45-s − 2·49-s + 16·59-s + 3·64-s − 24·71-s − 48·76-s − 16·79-s − 10·80-s + 81-s − 12·89-s + 32·95-s − 3·100-s − 8·101-s + 16·109-s − 24·116-s − 48·124-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s − 1/3·9-s + 5/4·16-s − 3.67·19-s − 1.34·20-s − 1/5·25-s − 1.48·29-s − 2.87·31-s − 1/2·36-s + 3.74·41-s + 0.298·45-s − 2/7·49-s + 2.08·59-s + 3/8·64-s − 2.84·71-s − 5.50·76-s − 1.80·79-s − 1.11·80-s + 1/9·81-s − 1.27·89-s + 3.28·95-s − 0.299·100-s − 0.796·101-s + 1.53·109-s − 2.22·116-s − 4.31·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9186475045\)
\(L(\frac12)\) \(\approx\) \(0.9186475045\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.406611522312070419945835214748, −8.892504296088618306004132580530, −8.620159437641553701178842379971, −8.392466054178223268517297026847, −7.55236634759414814958850890845, −7.48287639462489951606624654433, −7.32171732666136781469399155876, −6.71286856195478886250858434062, −6.23735372577332937544338302181, −5.82945191127033184202905508434, −5.80353173378479176327032114960, −5.01660817728515259090248791062, −4.14916561742667470191205768069, −4.05801419980960362158142521448, −3.83462340453073112989992800959, −2.83881055560240536008856200296, −2.58569693804047336298394069932, −1.90800047587759104968312137035, −1.72737423349988690003199800997, −0.31822386352142298968216031437, 0.31822386352142298968216031437, 1.72737423349988690003199800997, 1.90800047587759104968312137035, 2.58569693804047336298394069932, 2.83881055560240536008856200296, 3.83462340453073112989992800959, 4.05801419980960362158142521448, 4.14916561742667470191205768069, 5.01660817728515259090248791062, 5.80353173378479176327032114960, 5.82945191127033184202905508434, 6.23735372577332937544338302181, 6.71286856195478886250858434062, 7.32171732666136781469399155876, 7.48287639462489951606624654433, 7.55236634759414814958850890845, 8.392466054178223268517297026847, 8.620159437641553701178842379971, 8.892504296088618306004132580530, 9.406611522312070419945835214748

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