Properties

Degree 1
Conductor 83
Sign $-0.917 - 0.398i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.917 - 0.398i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (55, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.917 - 0.398i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2859344506 - 1.376320621i$
$L(\frac12,\chi)$  $\approx$  $0.2859344506 - 1.376320621i$
$L(\chi,1)$  $\approx$  0.6557267517 - 0.7355202488i
$L(1,\chi)$  $\approx$  0.6557267517 - 0.7355202488i

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

−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