Properties

Label 2-1083-1.1-c1-0-26
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 3·7-s + 9-s + 2·10-s − 3·11-s − 2·12-s + 6·13-s + 6·14-s − 15-s − 4·16-s + 3·17-s + 2·18-s + 2·20-s − 3·21-s − 6·22-s + 4·23-s − 4·25-s + 12·26-s − 27-s + 6·28-s + 10·29-s − 2·30-s − 2·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.632·10-s − 0.904·11-s − 0.577·12-s + 1.66·13-s + 1.60·14-s − 0.258·15-s − 16-s + 0.727·17-s + 0.471·18-s + 0.447·20-s − 0.654·21-s − 1.27·22-s + 0.834·23-s − 4/5·25-s + 2.35·26-s − 0.192·27-s + 1.13·28-s + 1.85·29-s − 0.365·30-s − 0.359·31-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: \(0\)
Selberg data: \((2,\ 1083,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.401675715\)
\(L(\frac12)\) \(\approx\) \(3.401675715\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−10.29447810506539493688558478858, −8.939322066982885493110877308795, −8.150338074826865833544092224319, −7.06801910593054313669065331224, −6.02567308265222601671827257647, −5.52130085797761183807247002982, −4.80392893944635697455333037352, −3.90431057417865746907593992810, −2.76056898471702515117641585129, −1.37927155401612470737240964655, 1.37927155401612470737240964655, 2.76056898471702515117641585129, 3.90431057417865746907593992810, 4.80392893944635697455333037352, 5.52130085797761183807247002982, 6.02567308265222601671827257647, 7.06801910593054313669065331224, 8.150338074826865833544092224319, 8.939322066982885493110877308795, 10.29447810506539493688558478858

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