Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 3·9-s + 8·11-s − 4·16-s − 6·18-s − 4·19-s + 16·22-s − 25-s − 8·32-s − 6·36-s − 8·38-s − 12·43-s + 16·44-s − 6·49-s − 2·50-s − 8·64-s − 12·67-s + 4·73-s − 8·76-s + 9·81-s + 4·83-s − 24·86-s + 16·89-s + 12·97-s − 12·98-s − 24·99-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 9-s + 2.41·11-s − 16-s − 1.41·18-s − 0.917·19-s + 3.41·22-s − 1/5·25-s − 1.41·32-s − 36-s − 1.29·38-s − 1.82·43-s + 2.41·44-s − 6/7·49-s − 0.282·50-s − 64-s − 1.46·67-s + 0.468·73-s − 0.917·76-s + 81-s + 0.439·83-s − 2.58·86-s + 1.69·89-s + 1.21·97-s − 1.21·98-s − 2.41·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{14400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 14400,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.068119899$
$L(\frac12)$  $\approx$  $2.068119899$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 + T^{2} \)
good7$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$V_4$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
31$V_4$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
37$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$V_4$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$V_4$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$V_4$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−11.47159085039242647938727597758, −10.96833487483284780227684094142, −10.15783307487823623866457104926, −9.412420274901051274202367362563, −8.904331528250912740116200744293, −8.587424534265571081920354606370, −7.67229869029425305034790087465, −6.66950658606595253187178805646, −6.45266453528203597964967881263, −5.95403439211185749520437525915, −5.08935044854913864128240694107, −4.43029119482209179710341171197, −3.73053720442232691090099083869, −3.18056876531371060126262025850, −1.92321555608603079637936047188, 1.92321555608603079637936047188, 3.18056876531371060126262025850, 3.73053720442232691090099083869, 4.43029119482209179710341171197, 5.08935044854913864128240694107, 5.95403439211185749520437525915, 6.45266453528203597964967881263, 6.66950658606595253187178805646, 7.67229869029425305034790087465, 8.587424534265571081920354606370, 8.904331528250912740116200744293, 9.412420274901051274202367362563, 10.15783307487823623866457104926, 10.96833487483284780227684094142, 11.47159085039242647938727597758

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