Properties

Label 4-720-1.1-c1e2-0-1
Degree $4$
Conductor $720$
Sign $1$
Analytic cond. $0.0459078$
Root an. cond. $0.462883$
Motivic weight $1$
Arithmetic yes
Rational yes
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

Degree: \(4\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.0459078\)
Root analytic conductor: \(0.462883\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 720,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3011887029\)
\(L(\frac12)\) \(\approx\) \(0.3011887029\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.8934316645, −19.1626037497, −19.1551537949, −18.6983747433, −17.6707519500, −16.9011141517, −16.6272610009, −16.0225385141, −15.4732516475, −14.9691010860, −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, 14.9691010860, 15.4732516475, 16.0225385141, 16.6272610009, 16.9011141517, 17.6707519500, 18.6983747433, 19.1551537949, 19.1626037497, 19.8934316645

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