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Properties

Label 4-1503-1.1-c1e2-0-0
Degree $4$
Conductor $1503$
Sign $-1$
Analytic cond. $0.0958325$
Root an. cond. $0.556388$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3-s + 4·4-s − 2·5-s + 3·6-s − 4·7-s − 3·8-s + 6·10-s − 4·12-s − 5·13-s + 12·14-s + 2·15-s + 3·16-s − 2·19-s − 8·20-s + 4·21-s + 23-s + 3·24-s + 15·26-s + 27-s − 16·28-s − 6·30-s + 4·31-s − 6·32-s + 8·35-s + 7·37-s + 6·38-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 2·4-s − 0.894·5-s + 1.22·6-s − 1.51·7-s − 1.06·8-s + 1.89·10-s − 1.15·12-s − 1.38·13-s + 3.20·14-s + 0.516·15-s + 3/4·16-s − 0.458·19-s − 1.78·20-s + 0.872·21-s + 0.208·23-s + 0.612·24-s + 2.94·26-s + 0.192·27-s − 3.02·28-s − 1.09·30-s + 0.718·31-s − 1.06·32-s + 1.35·35-s + 1.15·37-s + 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-1$
Analytic conductor: \(0.0958325\)
Root analytic conductor: \(0.556388\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1503,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 + T + T^{2} \)
167$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 19 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 24 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 9 T + 42 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 7 T - 16 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 8 T + 19 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−19.3547444054, −19.1030142671, −18.4881151360, −18.0531466655, −17.2411888544, −17.1348149136, −16.4473210857, −16.1826104829, −15.4186055231, −14.9570577763, −14.1142878547, −12.8880490467, −12.7381535490, −11.7971336069, −11.4409493233, −10.4061598711, −10.0423530573, −9.51264277468, −8.93211479480, −8.10247403315, −7.54929389557, −6.78086132143, −5.99753461842, −4.58011372366, −3.07479704531, 0, 3.07479704531, 4.58011372366, 5.99753461842, 6.78086132143, 7.54929389557, 8.10247403315, 8.93211479480, 9.51264277468, 10.0423530573, 10.4061598711, 11.4409493233, 11.7971336069, 12.7381535490, 12.8880490467, 14.1142878547, 14.9570577763, 15.4186055231, 16.1826104829, 16.4473210857, 17.1348149136, 17.2411888544, 18.0531466655, 18.4881151360, 19.1030142671, 19.3547444054

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