Properties

Label 4-1083-1.1-c1e2-0-0
Degree $4$
Conductor $1083$
Sign $1$
Analytic cond. $0.0690530$
Root an. cond. $0.512620$
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  − 2·2-s − 3-s + 4·5-s + 2·6-s + 2·7-s + 4·8-s − 8·10-s − 10·13-s − 4·14-s − 4·15-s − 4·16-s − 2·21-s + 4·23-s − 4·24-s + 3·25-s + 20·26-s + 4·27-s − 4·29-s + 8·30-s − 2·31-s + 8·35-s + 10·37-s + 10·39-s + 16·40-s − 14·41-s + 4·42-s − 2·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 1.78·5-s + 0.816·6-s + 0.755·7-s + 1.41·8-s − 2.52·10-s − 2.77·13-s − 1.06·14-s − 1.03·15-s − 16-s − 0.436·21-s + 0.834·23-s − 0.816·24-s + 3/5·25-s + 3.92·26-s + 0.769·27-s − 0.742·29-s + 1.46·30-s − 0.359·31-s + 1.35·35-s + 1.64·37-s + 1.60·39-s + 2.52·40-s − 2.18·41-s + 0.617·42-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.0690530\)
Root analytic conductor: \(0.512620\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1083} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1083,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2658373753\)
\(L(\frac12)\) \(\approx\) \(0.2658373753\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−19.6970630576, −19.1198479470, −18.4071259772, −18.1213987035, −17.4264315599, −17.4138610863, −16.7894756183, −16.7022471733, −15.2642565452, −14.5011418501, −14.3586980933, −13.3514918073, −13.0483250430, −12.0629600164, −11.4300293091, −10.2432044592, −10.2023031391, −9.40058826160, −9.18016854907, −8.06279874926, −7.42319981131, −6.39084306103, −5.03912355415, −5.02993077790, −2.13307257724, 2.13307257724, 5.02993077790, 5.03912355415, 6.39084306103, 7.42319981131, 8.06279874926, 9.18016854907, 9.40058826160, 10.2023031391, 10.2432044592, 11.4300293091, 12.0629600164, 13.0483250430, 13.3514918073, 14.3586980933, 14.5011418501, 15.2642565452, 16.7022471733, 16.7894756183, 17.4138610863, 17.4264315599, 18.1213987035, 18.4071259772, 19.1198479470, 19.6970630576

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