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Properties

Label 1-83-83.6-r1-0-0
Degree $1$
Conductor $83$
Sign $0.961 - 0.275i$
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  + (0.771 − 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (0.973 + 0.227i)5-s + (0.997 − 0.0765i)6-s + (0.720 + 0.693i)7-s + (−0.477 − 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (−0.953 − 0.301i)13-s + (0.997 + 0.0765i)14-s + (0.665 + 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯
L(s)  = 1  + (0.771 − 0.636i)2-s + (0.817 + 0.575i)3-s + (0.190 − 0.981i)4-s + (0.973 + 0.227i)5-s + (0.997 − 0.0765i)6-s + (0.720 + 0.693i)7-s + (−0.477 − 0.878i)8-s + (0.338 + 0.941i)9-s + (0.896 − 0.443i)10-s + (−0.665 + 0.746i)11-s + (0.720 − 0.693i)12-s + (−0.953 − 0.301i)13-s + (0.997 + 0.0765i)14-s + (0.665 + 0.746i)15-s + (−0.927 − 0.373i)16-s + (0.606 − 0.795i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.705323914 - 0.5201633161i\)
\(L(\frac12)\) \(\approx\) \(3.705323914 - 0.5201633161i\)
\(L(1)\) \(\approx\) \(2.314609725 - 0.3136703441i\)
\(L(1)\) \(\approx\) \(2.314609725 - 0.3136703441i\)

Euler product

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

−30.73852198684283500856458474063, −29.83549641281594810801794848673, −29.10769488393176173283798854327, −26.92229273800294908942056040497, −26.167853659328190959927626925991, −25.13681134686273227636691040958, −24.222297169260868507935406441277, −23.66521568493497561675290717528, −21.91835406659753918284152696943, −21.03294827697213086837589650788, −20.2103105958882577504196307617, −18.49358608206732867506627574592, −17.40514170001999365459703961715, −16.38836867135334522115889812862, −14.6422527431121786980377562659, −14.11385655947735305942320392523, −13.19123236846154889798328491757, −12.102644264316956019416765677590, −10.18442111791458000076314682241, −8.507837295939345038707350223169, −7.62994402796137601144195648443, −6.28710632326313231577999360314, −4.935327437326465158786175855535, −3.28540613912011374444019740374, −1.76235476668178509465326616204, 2.05698022381148407598308827990, 2.802583063564380145021604065096, 4.70519559328460378385229996114, 5.53016797529081330489524859772, 7.56769581602420905797891658060, 9.464238280358735950690130940451, 10.01410813943556686175991145834, 11.464186302014350736641203856839, 12.90352135003952958561936406000, 13.9954135052514776799015933613, 14.80401333358102513021239629900, 15.71563622821724456511537385771, 17.70553476694481505388193428271, 18.80135755920828755042658648945, 20.17307771838898562582314156844, 20.97100916547195613228745469281, 21.74407763656696210797581045225, 22.60420417855768544163149040236, 24.35658550247963134666407053211, 25.06171477814813753416658640744, 26.24078812116103655622425568617, 27.63598612451540145791895088539, 28.48530256500043556203371574629, 29.78201906910213477307678563729, 30.65526552503910410904447654473

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