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Properties

Label 1-83-83.22-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.942 + 0.335i$
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  + (−0.338 + 0.941i)2-s + (0.953 − 0.301i)3-s + (−0.771 − 0.636i)4-s + (0.114 + 0.993i)5-s + (−0.0383 + 0.999i)6-s + (−0.927 + 0.373i)7-s + (0.859 − 0.511i)8-s + (0.817 − 0.575i)9-s + (−0.973 − 0.227i)10-s + (−0.409 + 0.912i)11-s + (−0.927 − 0.373i)12-s + (−0.988 + 0.152i)13-s + (−0.0383 − 0.999i)14-s + (0.409 + 0.912i)15-s + (0.190 + 0.981i)16-s + (0.896 + 0.443i)17-s + ⋯
L(s)  = 1  + (−0.338 + 0.941i)2-s + (0.953 − 0.301i)3-s + (−0.771 − 0.636i)4-s + (0.114 + 0.993i)5-s + (−0.0383 + 0.999i)6-s + (−0.927 + 0.373i)7-s + (0.859 − 0.511i)8-s + (0.817 − 0.575i)9-s + (−0.973 − 0.227i)10-s + (−0.409 + 0.912i)11-s + (−0.927 − 0.373i)12-s + (−0.988 + 0.152i)13-s + (−0.0383 − 0.999i)14-s + (0.409 + 0.912i)15-s + (0.190 + 0.981i)16-s + (0.896 + 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.942 + 0.335i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.942 + 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2054319489 + 1.190254503i\)
\(L(\frac12)\) \(\approx\) \(0.2054319489 + 1.190254503i\)
\(L(1)\) \(\approx\) \(0.7487124379 + 0.6216528729i\)
\(L(1)\) \(\approx\) \(0.7487124379 + 0.6216528729i\)

Euler product

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

−29.917075497179143199467194701169, −29.20895734654809184420839739760, −28.02338202928300962615206456167, −26.99884538616056828867924592561, −26.1380904308772879192604061786, −25.08757545425279191730254142739, −23.74399238225084905972325219465, −22.11975813595757401783160833663, −21.28241342032494401826699114617, −20.19847520843336288095336318327, −19.62039085992978248881394139109, −18.56810307905960729588629973343, −16.85125933418712578114791317428, −16.12374953452094107491350494819, −14.25945996145130735511719602716, −13.16183710909186577588845010411, −12.47251302441959778305407544645, −10.63355056831289931898801841293, −9.57044165061675663006677409384, −8.77779908473655942906678848288, −7.54946994088406184820152657855, −5.040904144863411659504123420684, −3.68195607517336881469598888602, −2.47530682502127982411097475803, −0.55936316669609916749149815098, 2.19479059554689931447477950097, 3.80249320996361749950640278177, 5.87707275609912843029002129716, 7.109757934702316860148338292658, 7.87472444957390281111758213989, 9.58702640471754235522744967629, 10.037587561614312583699055263601, 12.46111395812023878786051060160, 13.62889756969512514395915531515, 14.87471955575087262193989388386, 15.233891101979373668149320898826, 16.81853935237721671401760717255, 18.23493610823453946590154596457, 18.93519318659383753797579843391, 19.80219894852250701652733005996, 21.58416798772511681138424694387, 22.748744616397592033008427255072, 23.73034130137163489148128220469, 25.20902148342774716097765883185, 25.64323970622541179204305545051, 26.4469889714681568591192456816, 27.50558318152478705425936476539, 29.00374947817284518582287366645, 30.14327149101428250535783891398, 31.46664790144514898484356265069

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