Properties

Degree $4$
Conductor $2433600$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 3·9-s + 8·11-s − 2·12-s − 16-s + 4·17-s − 3·18-s − 8·19-s − 8·22-s + 6·24-s + 25-s + 4·27-s − 5·32-s + 16·33-s − 4·34-s − 3·36-s + 8·38-s − 12·41-s − 8·43-s − 8·44-s − 2·48-s − 14·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 9-s + 2.41·11-s − 0.577·12-s − 1/4·16-s + 0.970·17-s − 0.707·18-s − 1.83·19-s − 1.70·22-s + 1.22·24-s + 1/5·25-s + 0.769·27-s − 0.883·32-s + 2.78·33-s − 0.685·34-s − 1/2·36-s + 1.29·38-s − 1.87·41-s − 1.21·43-s − 1.20·44-s − 0.288·48-s − 2·49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2433600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2433600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2433600,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−7.66093767427808108694408525270, −7.26587220658258987619659124673, −6.48654966834348628333399350819, −6.38045586811398028783284364586, −6.12557729231289131795228792800, −4.88964690901686673600763522174, −4.88823595164803979266926983864, −4.35289952326547862812898780702, −3.68764508509381599036667118399, −3.56799192480000330057521215275, −3.04823786935056050267442941329, −2.05348697617991381387369600911, −1.53827946418072491307494025034, −1.32104408130134713650580275933, 0, 1.32104408130134713650580275933, 1.53827946418072491307494025034, 2.05348697617991381387369600911, 3.04823786935056050267442941329, 3.56799192480000330057521215275, 3.68764508509381599036667118399, 4.35289952326547862812898780702, 4.88823595164803979266926983864, 4.88964690901686673600763522174, 6.12557729231289131795228792800, 6.38045586811398028783284364586, 6.48654966834348628333399350819, 7.26587220658258987619659124673, 7.66093767427808108694408525270

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