Properties

Label 4-156800-1.1-c1e2-0-7
Degree $4$
Conductor $156800$
Sign $-1$
Analytic cond. $9.99770$
Root an. cond. $1.77817$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s + 4-s + 4·6-s − 8-s + 6·9-s − 4·12-s − 8·13-s + 16-s − 6·18-s + 4·24-s − 5·25-s + 8·26-s + 4·27-s − 8·31-s − 32-s + 6·36-s + 4·37-s + 32·39-s + 12·41-s + 16·43-s − 4·48-s + 49-s + 5·50-s − 8·52-s + 12·53-s − 4·54-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.15·12-s − 2.21·13-s + 1/4·16-s − 1.41·18-s + 0.816·24-s − 25-s + 1.56·26-s + 0.769·27-s − 1.43·31-s − 0.176·32-s + 36-s + 0.657·37-s + 5.12·39-s + 1.87·41-s + 2.43·43-s − 0.577·48-s + 1/7·49-s + 0.707·50-s − 1.10·52-s + 1.64·53-s − 0.544·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(156800\)    =    \(2^{7} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(9.99770\)
Root analytic conductor: \(1.77817\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 156800,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
5$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.064899952938121589231451273163, −8.736817921425390383326382014904, −7.64673749340277573559389083829, −7.57571100088867902110310233811, −7.11642649742737663498250406032, −6.44599245093358620926386751475, −5.94966032900014960440899159541, −5.57928681742950427486583645839, −5.25236196288918634737499069062, −4.53924449636998865412903449029, −4.05250708127215343265869185182, −2.75791747804536479555350080645, −2.25326061715554950974013345200, −0.856225609282350254327766725168, 0, 0.856225609282350254327766725168, 2.25326061715554950974013345200, 2.75791747804536479555350080645, 4.05250708127215343265869185182, 4.53924449636998865412903449029, 5.25236196288918634737499069062, 5.57928681742950427486583645839, 5.94966032900014960440899159541, 6.44599245093358620926386751475, 7.11642649742737663498250406032, 7.57571100088867902110310233811, 7.64673749340277573559389083829, 8.736817921425390383326382014904, 9.064899952938121589231451273163

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