Properties

Label 2-1083-1.1-c1-0-46
Degree $2$
Conductor $1083$
Sign $-1$
Analytic cond. $8.64779$
Root an. cond. $2.94071$
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  + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s − 2·11-s + 12-s + 5·13-s + 14-s − 16-s − 4·17-s + 18-s − 21-s − 2·22-s − 4·23-s + 3·24-s − 5·25-s + 5·26-s − 27-s − 28-s − 8·29-s − 3·31-s + 5·32-s + 2·33-s − 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 25-s + 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.48·29-s − 0.538·31-s + 0.883·32-s + 0.348·33-s − 0.685·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1083\)    =    \(3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(8.64779\)
Root analytic conductor: \(2.94071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1083,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad3 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.490859406317181910865214053483, −8.595490268432358233091447782559, −7.88766191356537491665576331142, −6.62079287213053781803056519115, −5.84047896161452237997455144358, −5.18046014163099724265079503817, −4.20637327455860491087854991343, −3.48638822023419989452758004968, −1.86277933111965985175848882827, 0, 1.86277933111965985175848882827, 3.48638822023419989452758004968, 4.20637327455860491087854991343, 5.18046014163099724265079503817, 5.84047896161452237997455144358, 6.62079287213053781803056519115, 7.88766191356537491665576331142, 8.595490268432358233091447782559, 9.490859406317181910865214053483

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