Properties

Degree 1
Conductor 83
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{83} (82, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 83,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.514193680\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.514193680\)
\(L(\chi,1)\)  \(\approx\)  \(1.034503778\)
\(L(1,\chi)\)  \(\approx\)  \(1.034503778\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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