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Properties

Label 1-83-83.11-r0-0-0
Degree $1$
Conductor $83$
Sign $0.110 + 0.993i$
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  + (−0.264 + 0.964i)2-s + (0.896 + 0.443i)3-s + (−0.859 − 0.511i)4-s + (0.817 − 0.575i)5-s + (−0.665 + 0.746i)6-s + (−0.543 + 0.839i)7-s + (0.720 − 0.693i)8-s + (0.606 + 0.795i)9-s + (0.338 + 0.941i)10-s + (0.988 + 0.152i)11-s + (−0.543 − 0.839i)12-s + (−0.973 − 0.227i)13-s + (−0.665 − 0.746i)14-s + (0.988 − 0.152i)15-s + (0.477 + 0.878i)16-s + (−0.771 + 0.636i)17-s + ⋯
L(s)  = 1  + (−0.264 + 0.964i)2-s + (0.896 + 0.443i)3-s + (−0.859 − 0.511i)4-s + (0.817 − 0.575i)5-s + (−0.665 + 0.746i)6-s + (−0.543 + 0.839i)7-s + (0.720 − 0.693i)8-s + (0.606 + 0.795i)9-s + (0.338 + 0.941i)10-s + (0.988 + 0.152i)11-s + (−0.543 − 0.839i)12-s + (−0.973 − 0.227i)13-s + (−0.665 − 0.746i)14-s + (0.988 − 0.152i)15-s + (0.477 + 0.878i)16-s + (−0.771 + 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.110 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8272514179 + 0.7402696205i\)
\(L(\frac12)\) \(\approx\) \(0.8272514179 + 0.7402696205i\)
\(L(1)\) \(\approx\) \(0.9876813389 + 0.5923542385i\)
\(L(1)\) \(\approx\) \(0.9876813389 + 0.5923542385i\)

Euler product

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

−30.23349498787291407307462524264, −29.69891811331788051091505471657, −29.029102493066292744009860276534, −27.21201697038739846317025409194, −26.4817078797961705151710245412, −25.61164321799407018058880405799, −24.414876284520624572893790761407, −22.74166233835319126463268733178, −21.91168068951241512071883658069, −20.63627801265905092169230089842, −19.78316653827055422029900883154, −18.92658721776966436290440349721, −17.8269406439776865766508442554, −16.76828668612312483642911340768, −14.5549973757824972121704260349, −13.8324797327684291691331295030, −12.91404188436052514969402201102, −11.51900157243408631633970864537, −9.86834280083972482822746422897, −9.485471283444910013789274785772, −7.79170590950574997109182778634, −6.53585357259614904378332740419, −4.12743098606249065695018973061, −2.90187044670249115916897179118, −1.62553865911495105582477057056, 2.173473203565586430042134098617, 4.25163962065369963403159610361, 5.55145163398802079192943972903, 6.91531460946387631832278232342, 8.602235019966608112314599621077, 9.218347831095779703467065395239, 10.12213751706785827444685591657, 12.57505199545611121177164101596, 13.65638786668493874162489136409, 14.73380150455148681405293160984, 15.63334781603482209327348295513, 16.7794294896382810362956074787, 17.80774117173996967296196668960, 19.24751172497369260711457211816, 20.06756075174487915835679757153, 21.86382229750459868943586730255, 22.16523972927309524218629466341, 24.30779633481576729028060810481, 24.83698235795432913635243342702, 25.70802648631954311648750643508, 26.565600921074225524258764475466, 27.83589600134007338609681033957, 28.58904045337261394086000186858, 30.27374776095882253291637471160, 31.66854388291532292847384522151

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