Properties

Degree 1
Conductor 83
Sign $-0.942 - 0.335i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $-0.942 - 0.335i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (34, \cdot )$
Sato-Tate  :  $\mu(82)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (1:\ ),\ -0.942 - 0.335i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2054319489 - 1.190254503i$
$L(\frac12,\chi)$  $\approx$  $0.2054319489 - 1.190254503i$
$L(\chi,1)$  $\approx$  0.7487124379 - 0.6216528729i
$L(1,\chi)$  $\approx$  0.7487124379 - 0.6216528729i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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