Properties

Degree 1
Conductor 83
Sign $-0.676 + 0.736i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.927 − 0.373i)2-s + (−0.771 − 0.636i)3-s + (0.720 − 0.693i)4-s + (−0.606 + 0.795i)5-s + (−0.953 − 0.301i)6-s + (−0.997 − 0.0765i)7-s + (0.409 − 0.912i)8-s + (0.190 + 0.981i)9-s + (−0.264 + 0.964i)10-s + (−0.973 − 0.227i)11-s + (−0.997 + 0.0765i)12-s + (−0.338 + 0.941i)13-s + (−0.953 + 0.301i)14-s + (0.973 − 0.227i)15-s + (0.0383 − 0.999i)16-s + (−0.859 − 0.511i)17-s + ⋯
L(s,χ)  = 1  + (0.927 − 0.373i)2-s + (−0.771 − 0.636i)3-s + (0.720 − 0.693i)4-s + (−0.606 + 0.795i)5-s + (−0.953 − 0.301i)6-s + (−0.997 − 0.0765i)7-s + (0.409 − 0.912i)8-s + (0.190 + 0.981i)9-s + (−0.264 + 0.964i)10-s + (−0.973 − 0.227i)11-s + (−0.997 + 0.0765i)12-s + (−0.338 + 0.941i)13-s + (−0.953 + 0.301i)14-s + (0.973 − 0.227i)15-s + (0.0383 − 0.999i)16-s + (−0.859 − 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.676 + 0.736i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.676 + 0.736i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.676 + 0.736i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (13, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.676 + 0.736i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.004525310709 + 0.01030424856i$
$L(\frac12,\chi)$  $\approx$  $0.004525310709 + 0.01030424856i$
$L(\chi,1)$  $\approx$  0.8008636187 - 0.2513461425i
$L(1,\chi)$  $\approx$  0.8008636187 - 0.2513461425i

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

−30.39275827466905665225160552976, −28.875518661733072838746046526287, −28.530056654362764467045150939991, −26.923138858229758044322600967094, −25.966157384804616810835987825193, −24.54235006057755803202086801920, −23.616667508813039100997551291968, −22.72029874692567089622398375809, −21.94394821686488230776080523638, −20.65572209911645175517825172078, −19.84068499501284924398743226292, −17.79955557914854940999023281054, −16.60910937246566373992557887607, −15.7673938058055701665296038084, −15.22334821775719231349316175475, −13.137460616115289302192994381076, −12.53466622778482905178389875227, −11.33069287269308130666142614107, −9.97218771832663115482309778349, −8.2490872946366121741796005648, −6.71048915209195467838085575907, −5.409444007093348524751937262010, −4.48589149979575830103196629107, −3.10986286708595572490038398750, −0.003985543276669002610060327426, 2.2711668488360289599099663516, 3.71228728844445365079825014164, 5.368373825380238687749351000728, 6.61810990555475699139748422235, 7.418304614122189665643231428908, 9.98803234277266684925456182510, 11.11651377714764162519511807069, 11.97355927942434059614048884774, 13.100358294501758257280589190705, 14.07185514356320905109468037960, 15.66396452488934289302663205848, 16.38073614506400909913764962909, 18.34193522149268006028761254330, 19.05300148117370474214432828133, 20.083588141000161052147065398305, 21.829123452587472752188130469220, 22.45815523252086520546199584289, 23.42522550330181278573448051563, 24.06132866878382711321738948722, 25.48290873513170488318585701844, 26.78741903002504026383260647380, 28.33921337084487979516724409309, 29.220525294330690039370798948790, 29.76536896786848261325411028390, 31.17271456008501882462575460815

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