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Properties

Label 1-83-83.75-r0-0-0
Degree $1$
Conductor $83$
Sign $0.653 + 0.756i$
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  + (−0.973 − 0.227i)2-s + (−0.665 − 0.746i)3-s + (0.896 + 0.443i)4-s + (−0.997 + 0.0765i)5-s + (0.477 + 0.878i)6-s + (−0.264 + 0.964i)7-s + (−0.771 − 0.636i)8-s + (−0.114 + 0.993i)9-s + (0.988 + 0.152i)10-s + (0.720 − 0.693i)11-s + (−0.264 − 0.964i)12-s + (−0.409 + 0.912i)13-s + (0.477 − 0.878i)14-s + (0.720 + 0.693i)15-s + (0.606 + 0.795i)16-s + (0.953 + 0.301i)17-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)2-s + (−0.665 − 0.746i)3-s + (0.896 + 0.443i)4-s + (−0.997 + 0.0765i)5-s + (0.477 + 0.878i)6-s + (−0.264 + 0.964i)7-s + (−0.771 − 0.636i)8-s + (−0.114 + 0.993i)9-s + (0.988 + 0.152i)10-s + (0.720 − 0.693i)11-s + (−0.264 − 0.964i)12-s + (−0.409 + 0.912i)13-s + (0.477 − 0.878i)14-s + (0.720 + 0.693i)15-s + (0.606 + 0.795i)16-s + (0.953 + 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.653 + 0.756i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3206713741 + 0.1466816901i\)
\(L(\frac12)\) \(\approx\) \(0.3206713741 + 0.1466816901i\)
\(L(1)\) \(\approx\) \(0.4623484270 + 0.006013992788i\)
\(L(1)\) \(\approx\) \(0.4623484270 + 0.006013992788i\)

Euler product

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

−30.39602987172319467327253378582, −29.463592721021620862044557247254, −28.17173148354799982758354902160, −27.52647308847130766739008675969, −26.790288039777698722156815966983, −25.79112776485855528155370339288, −24.28450484493601293304968822039, −23.24841259429634062482200797281, −22.39950348773170773624304598351, −20.44717826155418985355455600404, −20.102609338753289103877680371321, −18.684671879494799576656069280731, −17.23777442460925493531492444934, −16.71596233163445162193219442724, −15.54231688266930598364937794582, −14.73618149821254049924799388930, −12.38809540486115687896994339264, −11.30190731433427493550268515238, −10.31886285819950235248100657544, −9.299125769718590748026501352899, −7.6931475385074865835894517453, −6.71193011216705104661411586756, −4.95480332005893777195936463202, −3.45521642901165002800800674566, −0.605351119361380544802244295108, 1.57291146337548579947004652658, 3.39977884615385201744017191504, 5.76598123135776315351605121623, 7.02263654753986693156648237567, 8.08150762111099704142986907335, 9.30602837160087964254591020605, 11.03346035828797874294368241374, 11.864147755912714760828903098964, 12.5194015874412845775741670408, 14.620842941718927203024458295973, 16.16908779546248659990088907971, 16.747634875719346448932184146017, 18.22780889240700296093282776302, 19.10353848689684661505011410471, 19.49999567846000604292796316803, 21.32095529357437892063017819289, 22.414869176099932339592785352531, 23.793893072947046319154725301275, 24.66163792866707971741062473793, 25.68528680321523579254847974193, 27.1841315372003415056003299924, 27.80032936056286983027320255658, 28.85238115940790851401144225952, 29.68997675380571647483829496443, 30.781644192139301864270919207196

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