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Properties

Label 1-83-83.54-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.312 - 0.949i$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.606 − 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (−0.953 + 0.301i)5-s + (0.409 + 0.912i)6-s + (0.477 + 0.878i)7-s + (0.927 − 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (0.114 − 0.993i)13-s + (0.409 − 0.912i)14-s + (0.997 − 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯
L(s)  = 1  + (−0.606 − 0.795i)2-s + (−0.973 − 0.227i)3-s + (−0.264 + 0.964i)4-s + (−0.953 + 0.301i)5-s + (0.409 + 0.912i)6-s + (0.477 + 0.878i)7-s + (0.927 − 0.373i)8-s + (0.896 + 0.443i)9-s + (0.817 + 0.575i)10-s + (−0.997 − 0.0765i)11-s + (0.477 − 0.878i)12-s + (0.114 − 0.993i)13-s + (0.409 − 0.912i)14-s + (0.997 − 0.0765i)15-s + (−0.859 − 0.511i)16-s + (0.338 + 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.312 - 0.949i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.312 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2696014964 - 0.3724207740i\)
\(L(\frac12)\) \(\approx\) \(0.2696014964 - 0.3724207740i\)
\(L(1)\) \(\approx\) \(0.4512398175 - 0.1642167928i\)
\(L(1)\) \(\approx\) \(0.4512398175 - 0.1642167928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (-0.606 - 0.795i)T \)
3 \( 1 + (-0.973 - 0.227i)T \)
5 \( 1 + (-0.953 + 0.301i)T \)
7 \( 1 + (0.477 + 0.878i)T \)
11 \( 1 + (-0.997 - 0.0765i)T \)
13 \( 1 + (0.114 - 0.993i)T \)
17 \( 1 + (0.338 + 0.941i)T \)
19 \( 1 + (-0.720 + 0.693i)T \)
23 \( 1 + (0.190 - 0.981i)T \)
29 \( 1 + (0.0383 - 0.999i)T \)
31 \( 1 + (-0.927 - 0.373i)T \)
37 \( 1 + (0.896 - 0.443i)T \)
41 \( 1 + (0.606 - 0.795i)T \)
43 \( 1 + (0.771 - 0.636i)T \)
47 \( 1 + (0.665 - 0.746i)T \)
53 \( 1 + (0.665 + 0.746i)T \)
59 \( 1 + (0.988 - 0.152i)T \)
61 \( 1 + (-0.543 - 0.839i)T \)
67 \( 1 + (0.859 + 0.511i)T \)
71 \( 1 + (-0.477 + 0.878i)T \)
73 \( 1 + (-0.817 - 0.575i)T \)
79 \( 1 + (0.264 - 0.964i)T \)
89 \( 1 + (0.409 + 0.912i)T \)
97 \( 1 + (0.409 - 0.912i)T \)
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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

−31.13705008712019023541060765915, −29.50025306187977221590119142911, −28.53838684782812940712681795909, −27.50307889998755131483915038484, −26.96526502924026533720648851016, −25.84702643984628284141968951356, −24.02468960453349683907760333738, −23.71039441326847195919142669902, −22.92587812785220795475206637028, −21.22182556198270802018188767409, −19.91787258154667257586234901910, −18.65136919837688583660757840349, −17.66443625589300689638343098188, −16.5049973200990274537552953107, −15.981908986697002835799139664218, −14.72436624283839836591775094933, −13.16346809086069457176287896079, −11.44908900592514012375589329743, −10.70910448235224851590559815727, −9.27121455009531620090497131797, −7.70068898830542813802283493725, −6.92396554467860660421593532895, −5.19543397518839733313180293285, −4.31887983926217837028762641186, −0.98920508162690811190067625779, 0.402739836777993387404351904800, 2.35790393130318285949293550039, 4.12928182101539646902076064978, 5.70984355545620612387710236685, 7.58893814115842943170493301980, 8.390542974765176664705039539372, 10.39529628578485058374536384373, 11.020424631556517463526441265924, 12.23887926377529740287001650649, 12.82964052746539480673546943295, 15.09211355197617951370521815266, 16.1998447101818935177082053027, 17.45866944712261144357897145652, 18.52653459235147582386785206803, 19.003330603286973691522100174229, 20.5784953446243571999503693023, 21.64924736297250727811017383270, 22.68146745547304185702929886528, 23.62410972240772026797741719919, 25.00040357214370225868850897381, 26.43928137990570164119491422313, 27.58441598886798868987023447269, 28.038763396921737156029455086112, 29.07223876391406045145628809237, 30.25576069853478239275502130759

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