Properties

Label 1-83-83.21-r0-0-0
Degree $1$
Conductor $83$
Sign $0.759 + 0.651i$
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  + (0.988 − 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (−0.543 + 0.839i)5-s + (0.190 + 0.981i)6-s + (0.338 − 0.941i)7-s + (0.896 − 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (0.720 + 0.693i)13-s + (0.190 − 0.981i)14-s + (−0.859 − 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯
L(s)  = 1  + (0.988 − 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (−0.543 + 0.839i)5-s + (0.190 + 0.981i)6-s + (0.338 − 0.941i)7-s + (0.896 − 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (0.720 + 0.693i)13-s + (0.190 − 0.981i)14-s + (−0.859 − 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.759 + 0.651i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.759 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479267430 + 0.5474772023i\)
\(L(\frac12)\) \(\approx\) \(1.479267430 + 0.5474772023i\)
\(L(1)\) \(\approx\) \(1.560063876 + 0.3738821907i\)
\(L(1)\) \(\approx\) \(1.560063876 + 0.3738821907i\)

Euler product

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

−31.19340969297100560320933123460, −29.91391273502495043625123777489, −28.80421300746352699700910698370, −28.0162496745319578590580106467, −26.07505337034195494755322004249, −24.944009924188600804350173726215, −24.24976789086925166564549953114, −23.54414925017461772419075773759, −22.38127156819892527284869312715, −21.00122006836674972040037177318, −20.138261121018011261088881692484, −18.91132897722617546963798149416, −17.65545760466743444488910111357, −16.16002044427349629991231694899, −15.30169193709993051623217769849, −13.85516060949972597800755034257, −12.836156570257773594322270540131, −12.12974177657787642140341701147, −11.03873717133513308462856324693, −8.4491633877780824062597268080, −7.85442576254966617491327890326, −6.06997908512940945938670466691, −5.234290068822049768744203035, −3.38742415724186439137773878052, −1.80443109893056413378980024669, 2.6550867649065598736766105924, 3.95425244162488348585409335152, 4.807662098246490972963196481307, 6.54063113680052725115699522046, 7.806380734519246146777197841855, 9.93429798300219437969562612003, 10.94269341381814825082821324500, 11.6201126265342782763873230237, 13.57332760651572414932579921429, 14.338928013808495299689105407775, 15.56818561812585859507644357247, 16.1229718767728365026191440831, 17.831741893037537287475431308747, 19.5423524813231090520027150669, 20.48817245798705809916531996082, 21.306490305998828502424411268350, 22.599427271593828626169547213831, 23.108250077917351051955271343145, 24.23456832864036386869331664207, 26.084721715258716841797297221728, 26.43350533052393000321295354348, 27.92556048743298247804539581506, 28.94870123287652824485674636766, 30.32596477958914174997634200691, 31.01499119356857955299324003106

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