Properties

Degree 1
Conductor 83
Sign $0.207 - 0.978i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.207 - 0.978i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (67, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ 0.207 - 0.978i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5214907199 - 0.4223992529i$
$L(\frac12,\chi)$  $\approx$  $0.5214907199 - 0.4223992529i$
$L(\chi,1)$  $\approx$  0.5402534892 - 0.1569446243i
$L(1,\chi)$  $\approx$  0.5402534892 - 0.1569446243i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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