Properties

Label 2-83-1.1-c1-0-6
Degree $2$
Conductor $83$
Sign $-1$
Analytic cond. $0.662758$
Root an. cond. $0.814099$
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 − 2·5-s + 6-s − 3·7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 12-s − 6·13-s + 3·14-s + 2·15-s − 16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s + 3·21-s − 3·22-s − 4·23-s − 3·24-s − 25-s + 6·26-s + 5·27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(83\)
Sign: $-1$
Analytic conductor: \(0.662758\)
Root analytic conductor: \(0.814099\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 83,\ (\ :1/2),\ -1)\)

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)$
bad83 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.82349410001976328687202362359, −12.30534590659785910801180963406, −11.77890434965261413314394451050, −10.17676635448238491848838037425, −9.457827669916618767543831706200, −8.118927993931557797577222055683, −6.97450299304430271706246705192, −5.31327024180315445602831381888, −3.65475901244453246589839654026, 0, 3.65475901244453246589839654026, 5.31327024180315445602831381888, 6.97450299304430271706246705192, 8.118927993931557797577222055683, 9.457827669916618767543831706200, 10.17676635448238491848838037425, 11.77890434965261413314394451050, 12.30534590659785910801180963406, 13.82349410001976328687202362359

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