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Properties

Label 1-83-83.55-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.917 - 0.398i$
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  + (−0.720 − 0.693i)2-s + (0.190 − 0.981i)3-s + (0.0383 + 0.999i)4-s + (0.264 − 0.964i)5-s + (−0.817 + 0.575i)6-s + (0.988 − 0.152i)7-s + (0.665 − 0.746i)8-s + (−0.927 − 0.373i)9-s + (−0.859 + 0.511i)10-s + (0.896 − 0.443i)11-s + (0.988 + 0.152i)12-s + (0.771 − 0.636i)13-s + (−0.817 − 0.575i)14-s + (−0.896 − 0.443i)15-s + (−0.997 + 0.0765i)16-s + (0.477 − 0.878i)17-s + ⋯
L(s)  = 1  + (−0.720 − 0.693i)2-s + (0.190 − 0.981i)3-s + (0.0383 + 0.999i)4-s + (0.264 − 0.964i)5-s + (−0.817 + 0.575i)6-s + (0.988 − 0.152i)7-s + (0.665 − 0.746i)8-s + (−0.927 − 0.373i)9-s + (−0.859 + 0.511i)10-s + (0.896 − 0.443i)11-s + (0.988 + 0.152i)12-s + (0.771 − 0.636i)13-s + (−0.817 − 0.575i)14-s + (−0.896 − 0.443i)15-s + (−0.997 + 0.0765i)16-s + (0.477 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.917 - 0.398i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.917 - 0.398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2859344506 - 1.376320621i\)
\(L(\frac12)\) \(\approx\) \(0.2859344506 - 1.376320621i\)
\(L(1)\) \(\approx\) \(0.6557267517 - 0.7355202488i\)
\(L(1)\) \(\approx\) \(0.6557267517 - 0.7355202488i\)

Euler product

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

−31.09932156476368818786931233065, −30.11366761857663233951419550199, −28.3299660287791781506143433458, −27.812911358699719686113339849455, −26.633954532586687431512885365828, −26.04157321816436897650573947653, −25.03257129641972164176827232714, −23.68832727746851223497265517960, −22.48133674289628287678331642103, −21.42385553148890427944527637080, −20.17874090118287567641210402613, −18.949044696913181936548743415703, −17.78787777438188987067350832997, −16.91404887100648969930698453290, −15.58190148102541277872178079820, −14.62339045966411681296951387084, −14.12279343838083839134728606044, −11.36981897488835710665086496738, −10.626011887760212335827231834, −9.39478133638089856666968393128, −8.38629502625052400767558106065, −6.88506810264579936089376270189, −5.577965554891443489808384628152, −4.06576333104402680917694378844, −1.953779300765316683970829727888, 0.91411569517785315248774745329, 1.76159914906164494965284549698, 3.68339661287817208966461569886, 5.67432627902738515414659905240, 7.56437116984798145358578354480, 8.39164755363132457154942925886, 9.42723574291964363364215275801, 11.23206034673879110579627315224, 12.0621126531550075287737407945, 13.19494695831424417284988483679, 14.20850276595069394501164987569, 16.33627381525352000276474216451, 17.36832741810317980412294413306, 18.13760115149078027116112119327, 19.27565732569411954023999352886, 20.45467997455804031374334171844, 20.90028203465136466344998496031, 22.56927779452396031881430499650, 24.069146342605756204614341632264, 24.918061484501364431915262935342, 25.74011194530799927793114418415, 27.44219451345237254524444453302, 27.89791038738443284906132914414, 29.37993112475318600322960547882, 29.816502653071470564536760138509

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