Properties

Label 4-6336-1.1-c1e2-0-0
Degree $4$
Conductor $6336$
Sign $1$
Analytic cond. $0.403988$
Root an. cond. $0.797245$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 9-s + 3·11-s − 16-s − 4·17-s − 18-s + 8·19-s − 3·22-s + 2·25-s − 5·32-s + 4·34-s − 36-s − 8·38-s − 12·41-s + 16·43-s − 3·44-s − 6·49-s − 2·50-s − 16·59-s + 7·64-s − 8·67-s + 4·68-s + 3·72-s − 20·73-s − 8·76-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1/3·9-s + 0.904·11-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.83·19-s − 0.639·22-s + 2/5·25-s − 0.883·32-s + 0.685·34-s − 1/6·36-s − 1.29·38-s − 1.87·41-s + 2.43·43-s − 0.452·44-s − 6/7·49-s − 0.282·50-s − 2.08·59-s + 7/8·64-s − 0.977·67-s + 0.485·68-s + 0.353·72-s − 2.34·73-s − 0.917·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6336\)    =    \(2^{6} \cdot 3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.403988\)
Root analytic conductor: \(0.797245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6000926798\)
\(L(\frac12)\) \(\approx\) \(0.6000926798\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.93439334804382919546995949213, −11.34497384419177131756738933013, −10.76780466456946714024310733355, −10.11672963358121359601593885655, −9.599815135362881295639350509225, −9.001815261298361697894118677486, −8.762989465694654662219980133086, −7.74426207559655217519353396530, −7.36925836009317348052168501648, −6.62619420165972560196427270392, −5.74469881629040321086611789900, −4.82090950308367352918122060264, −4.22600620193989736003151376656, −3.17606698937092513279428625412, −1.45988613491069469905927812537, 1.45988613491069469905927812537, 3.17606698937092513279428625412, 4.22600620193989736003151376656, 4.82090950308367352918122060264, 5.74469881629040321086611789900, 6.62619420165972560196427270392, 7.36925836009317348052168501648, 7.74426207559655217519353396530, 8.762989465694654662219980133086, 9.001815261298361697894118677486, 9.599815135362881295639350509225, 10.11672963358121359601593885655, 10.76780466456946714024310733355, 11.34497384419177131756738933013, 11.93439334804382919546995949213

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