Properties

Label 4-432e2-1.1-c1e2-0-0
Degree $4$
Conductor $186624$
Sign $1$
Analytic cond. $11.8993$
Root an. cond. $1.85729$
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  − 3·5-s − 3·7-s − 3·11-s + 5·13-s − 9·23-s + 25-s − 3·29-s − 9·31-s + 9·35-s + 4·37-s + 9·41-s + 9·43-s − 3·47-s − 49-s + 9·55-s + 3·59-s + 61-s − 15·65-s + 15·67-s + 24·71-s − 4·73-s + 9·77-s − 15·79-s − 15·83-s − 15·91-s + 5·97-s + 9·101-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.13·7-s − 0.904·11-s + 1.38·13-s − 1.87·23-s + 1/5·25-s − 0.557·29-s − 1.61·31-s + 1.52·35-s + 0.657·37-s + 1.40·41-s + 1.37·43-s − 0.437·47-s − 1/7·49-s + 1.21·55-s + 0.390·59-s + 0.128·61-s − 1.86·65-s + 1.83·67-s + 2.84·71-s − 0.468·73-s + 1.02·77-s − 1.68·79-s − 1.64·83-s − 1.57·91-s + 0.507·97-s + 0.895·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(186624\)    =    \(2^{8} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(11.8993\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{432} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 186624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6528724186\)
\(L(\frac12)\) \(\approx\) \(0.6528724186\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.30666223470059285580297598618, −11.11891800362927334959354332949, −10.56410182976454161878133929011, −9.869661786687927129368596611314, −9.770132792621821597984250814948, −9.073722340683754912567250884987, −8.603698370434085266204053662306, −8.074955773744659062333654214197, −7.75339736374419552933253366361, −7.37297901985174399572602851810, −6.75892452099719041101830918755, −5.96937158313213942312223206075, −5.96855201868062485240538828118, −5.21610563014850485601102771283, −4.28492090065725344306778087852, −3.77370142722726638115860021834, −3.68434356292360931445639533549, −2.79531521174003500966884648026, −1.97753131598309101828180766124, −0.50416412224080305732476102584, 0.50416412224080305732476102584, 1.97753131598309101828180766124, 2.79531521174003500966884648026, 3.68434356292360931445639533549, 3.77370142722726638115860021834, 4.28492090065725344306778087852, 5.21610563014850485601102771283, 5.96855201868062485240538828118, 5.96937158313213942312223206075, 6.75892452099719041101830918755, 7.37297901985174399572602851810, 7.75339736374419552933253366361, 8.074955773744659062333654214197, 8.603698370434085266204053662306, 9.073722340683754912567250884987, 9.770132792621821597984250814948, 9.869661786687927129368596611314, 10.56410182976454161878133929011, 11.11891800362927334959354332949, 11.30666223470059285580297598618

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