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Properties

Label 1-83-83.69-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.0645 - 0.997i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
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  + (−0.771 − 0.636i)2-s + (0.817 − 0.575i)3-s + (0.190 + 0.981i)4-s + (−0.973 + 0.227i)5-s + (−0.997 − 0.0765i)6-s + (0.720 − 0.693i)7-s + (0.477 − 0.878i)8-s + (0.338 − 0.941i)9-s + (0.896 + 0.443i)10-s + (−0.665 − 0.746i)11-s + (0.720 + 0.693i)12-s + (0.953 − 0.301i)13-s + (−0.997 + 0.0765i)14-s + (−0.665 + 0.746i)15-s + (−0.927 + 0.373i)16-s + (0.606 + 0.795i)17-s + ⋯
L(s)  = 1  + (−0.771 − 0.636i)2-s + (0.817 − 0.575i)3-s + (0.190 + 0.981i)4-s + (−0.973 + 0.227i)5-s + (−0.997 − 0.0765i)6-s + (0.720 − 0.693i)7-s + (0.477 − 0.878i)8-s + (0.338 − 0.941i)9-s + (0.896 + 0.443i)10-s + (−0.665 − 0.746i)11-s + (0.720 + 0.693i)12-s + (0.953 − 0.301i)13-s + (−0.997 + 0.0765i)14-s + (−0.665 + 0.746i)15-s + (−0.927 + 0.373i)16-s + (0.606 + 0.795i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.0645 - 0.997i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.0645 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5404492542 - 0.5765310567i\)
\(L(\frac12)\) \(\approx\) \(0.5404492542 - 0.5765310567i\)
\(L(1)\) \(\approx\) \(0.7351142803 - 0.4339409430i\)
\(L(1)\) \(\approx\) \(0.7351142803 - 0.4339409430i\)

Euler product

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

−31.51259582748972382207341894966, −30.34993162123360137644741856881, −28.34883429329039807947639783260, −27.79467584797900263414094622658, −26.95599152008897346255270399909, −25.86354367415171597887746299165, −25.04279197707645886509001229703, −23.922119477579706650725252972204, −22.87437701091842031371038281364, −20.979233564380558902751089819039, −20.35375027940794017371277818812, −19.016526820168596982346764192870, −18.32065058073489683837575009459, −16.64022264872084879005427504653, −15.58402235977571685695054787282, −15.097242420613877348156061245954, −13.83491199022743676231646596021, −11.87827531391496604290957473788, −10.57720593668821375707684914795, −9.27910725509577834957624188582, −8.20971129741316575178340300702, −7.57963645016326257254526228638, −5.47639655605151990412474826587, −4.14126108379850092447182184289, −2.07848542871376719742207135157, 1.17190879687712240283423513421, 2.973686664479458292313737661155, 4.03795885934323933340347800975, 6.85084500881427559118740021086, 8.15992239695028848062290236993, 8.39826087608884361996726682740, 10.39576994221521739382051682563, 11.31039855846221792447250791875, 12.6267263215043341262858603326, 13.73526515774019187062021404123, 15.175489380129390896535918615629, 16.44668768364700217602060719663, 17.962236970453601189037212562323, 18.735300321954127572474237050509, 19.74017222806127386031568267923, 20.49924731622918439394024557052, 21.53125590678585656402250896372, 23.4044481378032851689027533005, 24.14564484486100646178896302466, 25.71028564881729022314322260692, 26.43712592615040918454516405577, 27.305766718189313065792614232593, 28.3819117634213436704788862182, 29.94849949264037791352566305117, 30.34349543716782678497302966952

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