Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 5^{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  − 2·5-s + 9-s + 8·11-s − 8·19-s − 25-s + 12·29-s + 16·31-s − 12·41-s − 2·45-s − 14·49-s − 16·55-s + 8·59-s − 4·61-s + 16·71-s − 16·79-s + 81-s − 12·89-s + 16·95-s + 8·99-s − 36·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s + 1/3·9-s + 2.41·11-s − 1.83·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s − 1.87·41-s − 0.298·45-s − 2·49-s − 2.15·55-s + 1.04·59-s − 0.512·61-s + 1.89·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 1.64·95-s + 0.804·99-s − 3.58·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 14400,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.039896770$
$L(\frac12)$  $\approx$  $1.039896770$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.50178790877005747304683976268, −10.63029344891647763107725042646, −9.964671915727173343349228861429, −9.703395692631468853448056916432, −8.738054065213463721777507480749, −8.362573092386573492350464456460, −8.098990694093691505068092710868, −6.88451958096732153826690211844, −6.46625691967373786218224646311, −6.42897107072744719896577758454, −4.93489655467566708723346193761, −4.25303028692796488061253338187, −3.96665802174365669665598466984, −2.86970638476716148313728818699, −1.38738099288330045270314903054, 1.38738099288330045270314903054, 2.86970638476716148313728818699, 3.96665802174365669665598466984, 4.25303028692796488061253338187, 4.93489655467566708723346193761, 6.42897107072744719896577758454, 6.46625691967373786218224646311, 6.88451958096732153826690211844, 8.098990694093691505068092710868, 8.362573092386573492350464456460, 8.738054065213463721777507480749, 9.703395692631468853448056916432, 9.964671915727173343349228861429, 10.63029344891647763107725042646, 11.50178790877005747304683976268

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