Properties

Degree 4
Conductor $ 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
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

\( d \)  =  \(4\)
\( N \)  =  \(352800\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{352800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 352800,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.013615498$
$L(\frac12)$  $\approx$  $1.013615498$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

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