Properties

Label 4-594e2-1.1-c1e2-0-11
Degree $4$
Conductor $352836$
Sign $1$
Analytic cond. $22.4971$
Root an. cond. $2.17786$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 3·11-s + 5·16-s + 6·22-s − 25-s − 12·29-s + 10·31-s + 6·32-s + 4·37-s + 12·41-s + 9·44-s − 13·49-s − 2·50-s − 24·58-s + 20·62-s + 7·64-s + 28·67-s + 8·74-s + 24·82-s + 6·83-s + 12·88-s − 2·97-s − 26·98-s − 3·100-s + 6·101-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.904·11-s + 5/4·16-s + 1.27·22-s − 1/5·25-s − 2.22·29-s + 1.79·31-s + 1.06·32-s + 0.657·37-s + 1.87·41-s + 1.35·44-s − 1.85·49-s − 0.282·50-s − 3.15·58-s + 2.54·62-s + 7/8·64-s + 3.42·67-s + 0.929·74-s + 2.65·82-s + 0.658·83-s + 1.27·88-s − 0.203·97-s − 2.62·98-s − 0.299·100-s + 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352836 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352836 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(352836\)    =    \(2^{2} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(22.4971\)
Root analytic conductor: \(2.17786\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 352836,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.991386533\)
\(L(\frac12)\) \(\approx\) \(4.991386533\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - 3 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.628140728035937440023020847412, −8.015255053006877536003075086198, −7.82723843094092867969941089213, −7.12468139474762615848955209063, −6.71508725988629390136097161445, −6.29574961437633563574123917046, −5.84315760589664083898483470884, −5.39074241880952165115421872247, −4.77709333112371596512170551343, −4.30456169908575776721292993281, −3.80538107085536458846182311961, −3.38549178369413487864105189864, −2.58138575780524151090389938489, −2.03246862806699813641002498927, −1.09285120951784850918046733268, 1.09285120951784850918046733268, 2.03246862806699813641002498927, 2.58138575780524151090389938489, 3.38549178369413487864105189864, 3.80538107085536458846182311961, 4.30456169908575776721292993281, 4.77709333112371596512170551343, 5.39074241880952165115421872247, 5.84315760589664083898483470884, 6.29574961437633563574123917046, 6.71508725988629390136097161445, 7.12468139474762615848955209063, 7.82723843094092867969941089213, 8.015255053006877536003075086198, 8.628140728035937440023020847412

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