Properties

Degree 1
Conductor 83
Sign $0.755 + 0.654i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.817 + 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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.755 + 0.654i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (7, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.755 + 0.654i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.606404037 + 0.5988402447i$
$L(\frac12,\chi)$  $\approx$  $1.606404037 + 0.5988402447i$
$L(\chi,1)$  $\approx$  1.660915388 + 0.4762132913i
$L(1,\chi)$  $\approx$  1.660915388 + 0.4762132913i

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

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

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