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

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.676 - 0.736i$
Motivic weight: \(0\)
Character: $\chi_{83} (32, \cdot )$
Sato-Tate group: $\mu(82)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.676 - 0.736i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.004525310709 + 0.01030424856i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.004525310709 + 0.01030424856i\)
\(L(\chi,1)\) \(\approx\) \(0.8008636187 + 0.2513461425i\)
\(L(1,\chi)\) \(\approx\) \(0.8008636187 + 0.2513461425i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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