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Properties

Label 1-83-83.13-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.676 + 0.736i$
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  + (0.927 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.676 + 0.736i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.676 + 0.736i)\)

Particular Values

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

Euler product

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

−30.39275827466905665225160552976, −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