Properties

Degree 1
Conductor 139
Sign $0.0220 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.613 − 0.789i)2-s + (0.934 + 0.356i)3-s + (−0.247 − 0.968i)4-s + (−0.974 − 0.225i)5-s + (0.854 − 0.519i)6-s + (0.0227 − 0.999i)7-s + (−0.917 − 0.398i)8-s + (0.746 + 0.665i)9-s + (−0.775 + 0.631i)10-s + (0.995 − 0.0909i)11-s + (0.113 − 0.993i)12-s + (−0.158 − 0.987i)13-s + (−0.775 − 0.631i)14-s + (−0.829 − 0.557i)15-s + (−0.877 + 0.480i)16-s + (−0.998 + 0.0455i)17-s + ⋯
L(s,χ)  = 1  + (0.613 − 0.789i)2-s + (0.934 + 0.356i)3-s + (−0.247 − 0.968i)4-s + (−0.974 − 0.225i)5-s + (0.854 − 0.519i)6-s + (0.0227 − 0.999i)7-s + (−0.917 − 0.398i)8-s + (0.746 + 0.665i)9-s + (−0.775 + 0.631i)10-s + (0.995 − 0.0909i)11-s + (0.113 − 0.993i)12-s + (−0.158 − 0.987i)13-s + (−0.775 − 0.631i)14-s + (−0.829 − 0.557i)15-s + (−0.877 + 0.480i)16-s + (−0.998 + 0.0455i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0220 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0220 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(139\)
\( \varepsilon \)  =  $0.0220 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{139} (66, \cdot )$
Sato-Tate  :  $\mu(69)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 139,\ (0:\ ),\ 0.0220 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.183108929 - 1.157330251i$
$L(\frac12,\chi)$  $\approx$  $1.183108929 - 1.157330251i$
$L(\chi,1)$  $\approx$  1.344998850 - 0.7740045991i
$L(1,\chi)$  $\approx$  1.344998850 - 0.7740045991i

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

−28.67535740518700840207629040906, −27.2038213513066117396369880358, −26.5392185349740926364126291731, −25.564982511192694039786171199562, −24.4481862844094716279598850337, −24.18991922122173424928216840393, −22.73497884056085319155606948684, −21.916174279466549054580133862042, −20.76556759709761761919466472808, −19.52362628546309762100123370816, −18.74073033615095850659852134184, −17.51530912085764777145253288190, −16.02486057133390953300313530275, −15.24497656215402088307392258827, −14.548987328366847843729321360855, −13.48442081287781002553142569006, −12.23935478538501889570221824311, −11.5738513571931808292562770986, −9.06716598095963644499118982645, −8.66439765831284473614265816391, −7.21673220565197077840570865292, −6.590799611629764108655680936638, −4.704526508318308448768904913482, −3.64255974772730722844402881859, −2.406918673354350986774859890781, 1.32313204146855587013628940129, 3.2119286903781422136598950040, 3.89893753736417838710646200397, 4.91973766496263408804480672408, 6.94513706293416047425673956784, 8.22257379711098391529556329777, 9.44182169111861356475392736546, 10.542915665235368230607857506826, 11.545203567674444302892238367843, 12.85005434276826650711552949160, 13.72204075235746834082761280307, 14.79652719632254402574108643427, 15.522702526334815352588532096925, 16.882727945635940751381267414842, 18.5962675841604447100742744405, 19.81860759387877161167960674923, 19.96831125544805065293136950694, 20.897278221675248600268377750698, 22.255432500346436795996577215911, 22.97241601118679808068016003732, 24.19150815038757067939582593742, 24.945201169861571709354157858740, 26.64063086244394280988413542684, 27.224027954289362881065249993, 27.9800753922635176086706105086

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