Properties

Degree 4
Conductor $ 2^{7} \cdot 3^{2} \cdot 5^{3} $
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 − 5-s − 2·6-s − 8-s + 3·9-s + 10-s + 2·12-s + 4·13-s − 2·15-s + 16-s − 3·18-s − 20-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 2·30-s + 16·31-s − 32-s + 3·36-s + 4·37-s + 8·39-s + 40-s − 12·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 9-s + 0.316·10-s + 0.577·12-s + 1.10·13-s − 0.516·15-s + 1/4·16-s − 0.707·18-s − 0.223·20-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.365·30-s + 2.87·31-s − 0.176·32-s + 1/2·36-s + 0.657·37-s + 1.28·39-s + 0.158·40-s − 1.87·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 144000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(144000\)    =    \(2^{7} \cdot 3^{2} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{144000} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 144000,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.776497565$
$L(\frac12)$  $\approx$  $1.776497565$
$L(\frac{3}{2})$   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$C_1$ \( 1 + T \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.305587122869086271308905422277, −8.760711413234799859506496164896, −8.286230572976717874608236616986, −7.941231322257043910223245152113, −7.80207005021400136120224531772, −6.87333704174627289182797723156, −6.42217617652666865799421983967, −6.23991034800022982622767523992, −5.03421875022587082259817541459, −4.69263356392122883726033030816, −3.79769619454667096438258605845, −3.36585804145949552210873743484, −2.77580512142874145203019878589, −1.92677095659654979130865508090, −1.02898483249670331178782515928, 1.02898483249670331178782515928, 1.92677095659654979130865508090, 2.77580512142874145203019878589, 3.36585804145949552210873743484, 3.79769619454667096438258605845, 4.69263356392122883726033030816, 5.03421875022587082259817541459, 6.23991034800022982622767523992, 6.42217617652666865799421983967, 6.87333704174627289182797723156, 7.80207005021400136120224531772, 7.941231322257043910223245152113, 8.286230572976717874608236616986, 8.760711413234799859506496164896, 9.305587122869086271308905422277

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