Properties

Degree $4$
Conductor $56448$
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 + 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 − 2·24-s − 6·25-s − 4·27-s + 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 + 49-s − 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-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 − 0.408·24-s − 6/5·25-s − 0.769·27-s + 0.176·32-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 + 1/7·49-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56448\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{56448} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 56448,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.02545149330824803619517802253, −9.563538518206812440434518911703, −8.356918966632091922574869118017, −8.186632228453794414425905803942, −7.68501515018855362022465153557, −6.84552480602986516155667793601, −6.60807660052202573549483918899, −5.76536999919480153066315644301, −5.33985014787602837985094112170, −5.11740288034969242991482089170, −4.30167637849672643861137972376, −3.62482887081886485478101246558, −2.66792336770788585023270622114, −1.88351369279783613545674963411, 0, 1.88351369279783613545674963411, 2.66792336770788585023270622114, 3.62482887081886485478101246558, 4.30167637849672643861137972376, 5.11740288034969242991482089170, 5.33985014787602837985094112170, 5.76536999919480153066315644301, 6.60807660052202573549483918899, 6.84552480602986516155667793601, 7.68501515018855362022465153557, 8.186632228453794414425905803942, 8.356918966632091922574869118017, 9.563538518206812440434518911703, 10.02545149330824803619517802253

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