Properties

Label 1-83-83.82-r1-0-0
Degree $1$
Conductor $83$
Sign $1$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $1$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{83} (82, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.514193680\)
\(L(\frac12)\) \(\approx\) \(1.514193680\)
\(L(1)\) \(\approx\) \(1.034503778\)
\(L(1)\) \(\approx\) \(1.034503778\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.42625828885605464404754053425, −29.8175414743446227156649901676, −28.00655730668914839263179214377, −27.19646908631341211815177959955, −26.763907261751893345420094554507, −25.27120794014394381586801708542, −24.60413059275266014089825643164, −23.52164458554992604146030289458, −21.52541122809908710073631738364, −20.574298407733945331186303160384, −19.482550901856165576555907252030, −19.0508802564173414066333622989, −17.563391217363846108557770649288, −16.41001684935351027898122473984, −14.963328146137910103109816042148, −14.59568381529952022151224491492, −12.409656206446733262351632968997, −11.39863095518579283183326661167, −9.9944528157185826666408088977, −8.68516345573448593133850598003, −7.92691097611334965494046296489, −6.91759279924790059190355476608, −4.43836820028830768840739500457, −2.85164903161242844229259076713, −1.22292048422529503187794249765, 1.22292048422529503187794249765, 2.85164903161242844229259076713, 4.43836820028830768840739500457, 6.91759279924790059190355476608, 7.92691097611334965494046296489, 8.68516345573448593133850598003, 9.9944528157185826666408088977, 11.39863095518579283183326661167, 12.409656206446733262351632968997, 14.59568381529952022151224491492, 14.963328146137910103109816042148, 16.41001684935351027898122473984, 17.563391217363846108557770649288, 19.0508802564173414066333622989, 19.482550901856165576555907252030, 20.574298407733945331186303160384, 21.52541122809908710073631738364, 23.52164458554992604146030289458, 24.60413059275266014089825643164, 25.27120794014394381586801708542, 26.763907261751893345420094554507, 27.19646908631341211815177959955, 28.00655730668914839263179214377, 29.8175414743446227156649901676, 30.42625828885605464404754053425

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