Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
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 + 4-s − 3·5-s − 4·7-s − 8-s + 9-s + 3·10-s + 4·11-s + 4·14-s + 16-s + 8·17-s − 18-s − 8·19-s − 3·20-s − 4·22-s − 8·23-s + 2·25-s − 4·28-s + 16·31-s − 32-s − 8·34-s + 12·35-s + 36-s + 8·37-s + 8·38-s + 3·40-s − 12·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 1.06·14-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 1.83·19-s − 0.670·20-s − 0.852·22-s − 1.66·23-s + 2/5·25-s − 0.755·28-s + 2.87·31-s − 0.176·32-s − 1.37·34-s + 2.02·35-s + 1/6·36-s + 1.31·37-s + 1.29·38-s + 0.474·40-s − 1.87·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{720} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 720,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3011887029$
$L(\frac12)$  $\approx$  $0.3011887029$
$L(\frac{3}{2})$   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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−19.8934316645, −19.1626037497, −19.1551537949, −18.6983747433, −17.67075195, −16.9011141517, −16.6272610009, −16.0225385141, −15.4732516475, −14.969101086, −14.1819882616, −13.3351916393, −12.3172568607, −12.1436052702, −11.5820474614, −10.3916050944, −9.96467191573, −9.30558712287, −8.09899069409, −7.94123132226, −6.42897107073, −6.42217617653, −4.25303028693, −3.36585804146, 3.36585804146, 4.25303028693, 6.42217617653, 6.42897107073, 7.94123132226, 8.09899069409, 9.30558712287, 9.96467191573, 10.3916050944, 11.5820474614, 12.1436052702, 12.3172568607, 13.3351916393, 14.1819882616, 14.969101086, 15.4732516475, 16.0225385141, 16.6272610009, 16.9011141517, 17.67075195, 18.6983747433, 19.1551537949, 19.1626037497, 19.8934316645

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