Properties

Label 4-5280-1.1-c1e2-0-3
Degree $4$
Conductor $5280$
Sign $-1$
Analytic cond. $0.336657$
Root an. cond. $0.761722$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 2·10-s − 7·11-s − 2·12-s + 4·15-s + 16-s − 2·20-s + 7·22-s − 23-s + 2·24-s − 2·25-s + 5·27-s − 2·29-s − 4·30-s + 31-s − 32-s + 14·33-s − 7·37-s + 2·40-s − 8·41-s − 2·43-s − 7·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 0.632·10-s − 2.11·11-s − 0.577·12-s + 1.03·15-s + 1/4·16-s − 0.447·20-s + 1.49·22-s − 0.208·23-s + 0.408·24-s − 2/5·25-s + 0.962·27-s − 0.371·29-s − 0.730·30-s + 0.179·31-s − 0.176·32-s + 2.43·33-s − 1.15·37-s + 0.316·40-s − 1.24·41-s − 0.304·43-s − 1.05·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5280\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(0.336657\)
Root analytic conductor: \(0.761722\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5280,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
good7$C_2^2$ \( 1 + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 28 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 5 T + 94 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.7308434271, −17.1482921924, −16.7281692270, −16.2569417737, −15.7129020660, −15.3879677112, −15.0215741520, −13.9688163449, −13.5113560193, −12.7994175777, −12.1627214288, −11.8575215326, −11.2494431828, −10.7250489311, −10.3599000291, −9.78701249545, −8.66478301902, −8.34339903722, −7.57211759043, −7.16265966585, −6.18712579402, −5.44996856635, −5.00831454295, −3.72597408133, −2.53064733255, 0, 2.53064733255, 3.72597408133, 5.00831454295, 5.44996856635, 6.18712579402, 7.16265966585, 7.57211759043, 8.34339903722, 8.66478301902, 9.78701249545, 10.3599000291, 10.7250489311, 11.2494431828, 11.8575215326, 12.1627214288, 12.7994175777, 13.5113560193, 13.9688163449, 15.0215741520, 15.3879677112, 15.7129020660, 16.2569417737, 16.7281692270, 17.1482921924, 17.7308434271

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