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Properties

Label 4-8788-1.1-c1e2-0-3
Degree $4$
Conductor $8788$
Sign $-1$
Analytic cond. $0.560330$
Root an. cond. $0.865189$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 6·5-s − 4·7-s − 5·9-s + 6·11-s + 13-s + 16-s − 6·17-s − 4·19-s − 6·20-s + 6·23-s + 17·25-s − 4·28-s + 6·29-s − 4·31-s + 24·35-s − 5·36-s − 10·37-s + 6·44-s + 30·45-s − 49-s + 52-s − 6·53-s − 36·55-s + 20·63-s + 64-s − 6·65-s + ⋯
L(s)  = 1  + 1/2·4-s − 2.68·5-s − 1.51·7-s − 5/3·9-s + 1.80·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.34·20-s + 1.25·23-s + 17/5·25-s − 0.755·28-s + 1.11·29-s − 0.718·31-s + 4.05·35-s − 5/6·36-s − 1.64·37-s + 0.904·44-s + 4.47·45-s − 1/7·49-s + 0.138·52-s − 0.824·53-s − 4.85·55-s + 2.51·63-s + 1/8·64-s − 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8788\)    =    \(2^{2} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(0.560330\)
Root analytic conductor: \(0.865189\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 8788,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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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

−16.7401598333, −16.3289342707, −16.1610422211, −15.3407982120, −15.2621223361, −14.7871828532, −14.1081230548, −13.5230676379, −12.6948580012, −12.1081086828, −12.0906172632, −11.2148699926, −11.1734099077, −10.6571590377, −9.25208611998, −9.14676247888, −8.28912505160, −8.14577120022, −6.89586188103, −6.76423132328, −6.28038385959, −4.97317601948, −3.91117219162, −3.64028761626, −2.87766967615, 0, 2.87766967615, 3.64028761626, 3.91117219162, 4.97317601948, 6.28038385959, 6.76423132328, 6.89586188103, 8.14577120022, 8.28912505160, 9.14676247888, 9.25208611998, 10.6571590377, 11.1734099077, 11.2148699926, 12.0906172632, 12.1081086828, 12.6948580012, 13.5230676379, 14.1081230548, 14.7871828532, 15.2621223361, 15.3407982120, 16.1610422211, 16.3289342707, 16.7401598333

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