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Properties

Label 1-83-83.14-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: $\chi_{83} (14, \cdot )$
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

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

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