Properties

Label 4-352800-1.1-c1e2-0-5
Degree $4$
Conductor $352800$
Sign $1$
Analytic cond. $22.4948$
Root an. cond. $2.17781$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s + 2·7-s − 8-s + 9-s − 2·12-s − 2·14-s + 16-s + 12·17-s − 18-s − 4·21-s + 2·24-s − 5·25-s + 4·27-s + 2·28-s − 32-s − 12·34-s + 36-s + 4·42-s + 16·43-s − 2·48-s + 3·49-s + 5·50-s − 24·51-s + 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.534·14-s + 1/4·16-s + 2.91·17-s − 0.235·18-s − 0.872·21-s + 0.408·24-s − 25-s + 0.769·27-s + 0.377·28-s − 0.176·32-s − 2.05·34-s + 1/6·36-s + 0.617·42-s + 2.43·43-s − 0.288·48-s + 3/7·49-s + 0.707·50-s − 3.36·51-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(352800\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(22.4948\)
Root analytic conductor: \(2.17781\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 352800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.013615498\)
\(L(\frac12)\) \(\approx\) \(1.013615498\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.641946436905462506660121810666, −8.245441784546572320334189108752, −7.61188392109476428586562825239, −7.57571100088867902110310233811, −7.00040208558644603028855906365, −6.24370967191199643899749739066, −5.77713278642514260434446052217, −5.57928681742950427486583645839, −5.14472978179797247228927582624, −4.37172349798127682503408367548, −3.78880803467157430054818673540, −3.08226175311992974929178698307, −2.33138830690794607760289984480, −1.34951115258711429409173467390, −0.77732160932427153121684324644, 0.77732160932427153121684324644, 1.34951115258711429409173467390, 2.33138830690794607760289984480, 3.08226175311992974929178698307, 3.78880803467157430054818673540, 4.37172349798127682503408367548, 5.14472978179797247228927582624, 5.57928681742950427486583645839, 5.77713278642514260434446052217, 6.24370967191199643899749739066, 7.00040208558644603028855906365, 7.57571100088867902110310233811, 7.61188392109476428586562825239, 8.245441784546572320334189108752, 8.641946436905462506660121810666

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