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Properties

Label 1-83-83.67-r1-0-0
Degree $1$
Conductor $83$
Sign $0.207 - 0.978i$
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  + (−0.953 − 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (0.409 − 0.912i)5-s + (0.927 + 0.373i)6-s + (−0.771 + 0.636i)7-s + (−0.606 − 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (−0.0383 + 0.999i)13-s + (0.927 − 0.373i)14-s + (−0.477 + 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (0.409 − 0.912i)5-s + (0.927 + 0.373i)6-s + (−0.771 + 0.636i)7-s + (−0.606 − 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (−0.0383 + 0.999i)13-s + (0.927 − 0.373i)14-s + (−0.477 + 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.207 - 0.978i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ 0.207 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5214907199 - 0.4223992529i\)
\(L(\frac12)\) \(\approx\) \(0.5214907199 - 0.4223992529i\)
\(L(1)\) \(\approx\) \(0.5402534892 - 0.1569446243i\)
\(L(1)\) \(\approx\) \(0.5402534892 - 0.1569446243i\)

Euler product

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

−30.17413471290198225064882049285, −29.5319059278143335098814920934, −28.83671557164166060104432630963, −27.44020475545992015235407723907, −26.796327298989783603092845138708, −25.71919599435791876080346179260, −24.58906767155544258418514522686, −23.29182937350957974307313738268, −22.45070931815638792211546045676, −21.2082765884861476877654897428, −19.58192657961988542780166884834, −18.70285529453618081111212620412, −17.59963198936634153138504753379, −16.83267579882565407154689108884, −15.79538599047624491150261752370, −14.51621332819216553521557127881, −12.8689818459981832429147583918, −11.18225401846422763571504918191, −10.514303837224200668317920795599, −9.56112724497581806605377462998, −7.66728883982159091342154761412, −6.47032472087934028426680418523, −5.78361473658806725662234527234, −3.33159196288241154000174477288, −1.10270933275422005389585728652, 0.603815269405966918012000804763, 2.18808543047482890473604427244, 4.55494750136350679457526876661, 6.15165129622978893308438545628, 7.224877822104844732509970635626, 9.22958016745486986968784684504, 9.59772432965945217617105038918, 11.329113127239333318966423445024, 12.201716444155691012451180947929, 13.123730463842873428004964777938, 15.51856308368753310974945871231, 16.47034186089919381000943358938, 17.224690462439805365305759628085, 18.27195200016829574262873849003, 19.33064330885067217736720705497, 20.613093405841794337532348782096, 21.65665089069604845118436544194, 22.69967450117833333107777834367, 24.3109957690767365607423024681, 25.025903529972993668696215343879, 26.23236155449751201322146748563, 27.611661003130990146075991390174, 28.47229304849330828346836817002, 28.84191736909576125925750615001, 29.90608254822692461105044911498

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