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Properties

Label 1-83-83.51-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.636 + 0.771i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
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  + (−0.927 − 0.373i)2-s + (−0.771 + 0.636i)3-s + (0.720 + 0.693i)4-s + (0.606 + 0.795i)5-s + (0.953 − 0.301i)6-s + (−0.997 + 0.0765i)7-s + (−0.409 − 0.912i)8-s + (0.190 − 0.981i)9-s + (−0.264 − 0.964i)10-s + (−0.973 + 0.227i)11-s + (−0.997 − 0.0765i)12-s + (0.338 + 0.941i)13-s + (0.953 + 0.301i)14-s + (−0.973 − 0.227i)15-s + (0.0383 + 0.999i)16-s + (−0.859 + 0.511i)17-s + ⋯
L(s)  = 1  + (−0.927 − 0.373i)2-s + (−0.771 + 0.636i)3-s + (0.720 + 0.693i)4-s + (0.606 + 0.795i)5-s + (0.953 − 0.301i)6-s + (−0.997 + 0.0765i)7-s + (−0.409 − 0.912i)8-s + (0.190 − 0.981i)9-s + (−0.264 − 0.964i)10-s + (−0.973 + 0.227i)11-s + (−0.997 − 0.0765i)12-s + (0.338 + 0.941i)13-s + (0.953 + 0.301i)14-s + (−0.973 − 0.227i)15-s + (0.0383 + 0.999i)16-s + (−0.859 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1395655610 + 0.2961536808i\)
\(L(\frac12)\) \(\approx\) \(0.1395655610 + 0.2961536808i\)
\(L(1)\) \(\approx\) \(0.4290377427 + 0.1779678811i\)
\(L(1)\) \(\approx\) \(0.4290377427 + 0.1779678811i\)

Euler product

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

−29.86287015426308750335077308077, −29.19289375368220639202873255473, −28.456229063862510700267940945299, −27.602747241634046704462167038343, −26.00303291637777831573555878178, −25.26742194394163552646472906016, −24.19523229935648978144661694032, −23.38393040982044983358244171339, −21.98998447705500044052237084847, −20.4349100832472529218277409828, −19.43445240049555074493156213420, −18.18285084556260820862250825717, −17.51233973890730169757394095262, −16.35863665008189740960098759083, −15.67764757761808922067983467297, −13.46895754175175939211194115028, −12.7280348015120992745755510288, −11.099627934354341584953591142817, −10.04800404017080898933040011763, −8.76177594200263524498136586874, −7.42114305138425822555260234777, −6.15082402727980563120950469376, −5.286486583785532850064042095500, −2.30733264236244426324179890285, −0.49247955502936551665283153510, 2.30749094216623236074405346240, 3.86066512872454327065158621798, 6.06499133117672629800623427945, 6.8630700930097377240896981508, 8.84082455305131539424987919648, 10.12644599233932225506421150185, 10.53230866096865025896000100136, 11.898540883735153246526910715725, 13.184893641942971366692403584850, 15.148950203207348205669970923984, 16.160966583335297164646870145523, 17.11513064191725916388497047496, 18.24454508405065422488142241416, 18.9869660539276592153031797210, 20.57658959705003146450338155251, 21.574479239140964479318156848940, 22.34414708363871600579653047708, 23.62156976170088858564258592339, 25.41259962894696291308721140374, 26.270656756919793070669502407778, 26.83392113206503491045591558105, 28.49279279684126304444214761197, 28.7697457278902857084129937348, 29.73488895638080390996271806847, 31.02060956014389234532276245519

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