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Properties

Label 1-83-83.12-r0-0-0
Degree $1$
Conductor $83$
Sign $0.755 - 0.654i$
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  + (0.817 − 0.575i)2-s + (0.988 − 0.152i)3-s + (0.338 − 0.941i)4-s + (−0.665 + 0.746i)5-s + (0.720 − 0.693i)6-s + (0.190 + 0.981i)7-s + (−0.264 − 0.964i)8-s + (0.953 − 0.301i)9-s + (−0.114 + 0.993i)10-s + (−0.543 − 0.839i)11-s + (0.190 − 0.981i)12-s + (−0.997 + 0.0765i)13-s + (0.720 + 0.693i)14-s + (−0.543 + 0.839i)15-s + (−0.771 − 0.636i)16-s + (−0.973 − 0.227i)17-s + ⋯
L(s)  = 1  + (0.817 − 0.575i)2-s + (0.988 − 0.152i)3-s + (0.338 − 0.941i)4-s + (−0.665 + 0.746i)5-s + (0.720 − 0.693i)6-s + (0.190 + 0.981i)7-s + (−0.264 − 0.964i)8-s + (0.953 − 0.301i)9-s + (−0.114 + 0.993i)10-s + (−0.543 − 0.839i)11-s + (0.190 − 0.981i)12-s + (−0.997 + 0.0765i)13-s + (0.720 + 0.693i)14-s + (−0.543 + 0.839i)15-s + (−0.771 − 0.636i)16-s + (−0.973 − 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606404037 - 0.5988402447i\)
\(L(\frac12)\) \(\approx\) \(1.606404037 - 0.5988402447i\)
\(L(1)\) \(\approx\) \(1.660915388 - 0.4762132913i\)
\(L(1)\) \(\approx\) \(1.660915388 - 0.4762132913i\)

Euler product

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

−31.10575560013631491777520844058, −30.500975280982021695405591739667, −29.16282194980081725547169080454, −27.41928061498418344599078487740, −26.54688409009728203235063096809, −25.61199800104541260670159356218, −24.362862421289086104155425837806, −23.82537743479075500551714973896, −22.55988522920961836775065312009, −21.14076100501172256582062625558, −20.31496576369768817416167483038, −19.59413643381068699602086778449, −17.54819857600465577844798185788, −16.4407456382097743375956160619, −15.321209522632486107601376096249, −14.56808071066310860933509717530, −13.17937266916094561478744951799, −12.58358587485413645918662167278, −10.78849231945760669652115146019, −9.02639293284335828213760153494, −7.81250137049156107897889060686, −7.07223669767342898281068580326, −4.69357534281228449677452942531, −4.21035036015498383768858494158, −2.4655743900662158335089987796, 2.298554407234016587694619423284, 3.114879968722991728526360986051, 4.57922961002327451266498291663, 6.32025703459080975085688608994, 7.76833185883968438315477245158, 9.19503205596130639092293421790, 10.65460201514527658726661621176, 11.82818277903975483645977108060, 12.95259644152609975716762536199, 14.18937361707437989004622838697, 15.06884943320311349269595800844, 15.75561257532472051621271706552, 18.286867514846157547480393323738, 19.14671701072319469001397980084, 19.790857073006949267473493238932, 21.29657850488300724996665011376, 21.83900084348000346238914727508, 23.24082620274372369547713898401, 24.32347206396817762131184599404, 25.15739523059037390461201220670, 26.65061820353041551462307974683, 27.475554880849083004371331998525, 29.03473160333210877116604587431, 29.922816784214501604017442596054, 31.12918772698569374879422653282

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