Properties

Degree 4
Conductor $ 2^{7} \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  − 3·9-s + 8·11-s − 4·13-s + 8·23-s + 25-s + 12·37-s + 8·47-s + 2·49-s − 8·59-s − 4·61-s − 12·73-s + 9·81-s − 32·83-s − 28·97-s − 24·99-s + 28·109-s + 12·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 32·143-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 9-s + 2.41·11-s − 1.10·13-s + 1.66·23-s + 1/5·25-s + 1.97·37-s + 1.16·47-s + 2/7·49-s − 1.04·59-s − 0.512·61-s − 1.40·73-s + 81-s − 3.51·83-s − 2.84·97-s − 2.41·99-s + 2.68·109-s + 1.10·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(28800\)    =    \(2^{7} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 28800,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.272179997$
$L(\frac12)$  $\approx$  $1.272179997$
$L(\frac{3}{2})$   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 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−10.65133305239162895186920528545, −9.851840183892789826086741161768, −9.401949169928688362704456671527, −9.022035638415746178025932660186, −8.635268153066941217007271839152, −7.88125929411027787932176182764, −7.00520502565783528594752431202, −6.97238544391138654785050907556, −5.99573481729811170443753568761, −5.70093888866808737514223874840, −4.61878022728753276110930411726, −4.28002313553101117805216295071, −3.26180587408034592610487247010, −2.60538869357449517223391665122, −1.23519091396578725398334381678, 1.23519091396578725398334381678, 2.60538869357449517223391665122, 3.26180587408034592610487247010, 4.28002313553101117805216295071, 4.61878022728753276110930411726, 5.70093888866808737514223874840, 5.99573481729811170443753568761, 6.97238544391138654785050907556, 7.00520502565783528594752431202, 7.88125929411027787932176182764, 8.635268153066941217007271839152, 9.022035638415746178025932660186, 9.401949169928688362704456671527, 9.851840183892789826086741161768, 10.65133305239162895186920528545

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