Properties

Label 2-83-1.1-c1-0-2
Degree $2$
Conductor $83$
Sign $1$
Analytic cond. $0.662758$
Root an. cond. $0.814099$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.429·2-s + 2.76·3-s − 1.81·4-s + 1.71·5-s − 1.18·6-s − 3.46·7-s + 1.63·8-s + 4.65·9-s − 0.735·10-s − 4.04·11-s − 5.02·12-s + 4.06·13-s + 1.48·14-s + 4.74·15-s + 2.92·16-s − 4.60·17-s − 2·18-s − 2.09·19-s − 3.11·20-s − 9.58·21-s + 1.73·22-s − 3.55·23-s + 4.53·24-s − 2.06·25-s − 1.74·26-s + 4.58·27-s + 6.29·28-s + ⋯
L(s)  = 1  − 0.303·2-s + 1.59·3-s − 0.907·4-s + 0.766·5-s − 0.485·6-s − 1.30·7-s + 0.579·8-s + 1.55·9-s − 0.232·10-s − 1.21·11-s − 1.45·12-s + 1.12·13-s + 0.397·14-s + 1.22·15-s + 0.731·16-s − 1.11·17-s − 0.471·18-s − 0.479·19-s − 0.695·20-s − 2.09·21-s + 0.369·22-s − 0.740·23-s + 0.925·24-s − 0.412·25-s − 0.342·26-s + 0.883·27-s + 1.18·28-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 83,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.081561962\)
\(L(\frac12)\) \(\approx\) \(1.081561962\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.429T + 2T^{2} \)
3 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 - 1.71T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 - 4.06T + 13T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 0.181T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 8.34T + 37T^{2} \)
41 \( 1 - 5.35T + 41T^{2} \)
43 \( 1 + 3.68T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 - 1.97T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + 4.50T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 0.572T + 79T^{2} \)
89 \( 1 + 4.48T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−13.89796330619988133600325778523, −13.40213766818186345208601826991, −12.92930577012930543185175492708, −10.39885260012894691259326947510, −9.632957247343016353934898793094, −8.813936391112972016593412793869, −7.914310921490220011058263009318, −6.14892696246517096812732182564, −4.09273863272552915947411507239, −2.61303096446109283914280089532, 2.61303096446109283914280089532, 4.09273863272552915947411507239, 6.14892696246517096812732182564, 7.914310921490220011058263009318, 8.813936391112972016593412793869, 9.632957247343016353934898793094, 10.39885260012894691259326947510, 12.92930577012930543185175492708, 13.40213766818186345208601826991, 13.89796330619988133600325778523

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