Properties

Degree 1
Conductor 13
Sign $0.252 + 0.967i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.252 + 0.967i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.252 + 0.967i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $0.252 + 0.967i$
motivic weight  =  \(0\)
character  :  $\chi_{13} (3, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 13,\ (0:\ ),\ 0.252 + 0.967i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3386662820 + 0.2615970463i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3386662820 + 0.2615970463i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5663299165 + 0.3150964447i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5663299165 + 0.3150964447i\)

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

−44.47903323940722017826958439666, −41.890649557271878583545590023119, −40.75102754054690218810039287064, −39.64551328334837142992430136246, −37.84387198040439541736000486903, −36.72655061378587091397240745626, −35.40886810623710413108843015950, −34.19316896008938726193432669118, −31.85201648014964384144879878463, −30.217328576786274410480730339152, −29.07833159170359986891297897720, −28.30140715237261539047320509061, −26.09891977749163645666677859565, −24.70097591492895478309299511489, −22.50218000396131880869484608578, −21.29072366419482670542059960211, −19.16577686946377440528872864956, −18.1774758218214111196391353749, −16.77349086567586963433913803006, −13.51073102822369922801862938712, −12.3257791686484817550992480946, −10.506730485058794664625749128515, −8.53434388229500193005541581860, −5.99433348263937573006936411548, −2.27313120101337092309675281122, 4.938590808457977145461225357659, 6.72931431168193311808562367956, 9.4141759092143440176101772470, 10.457720324996851347033210348715, 13.617240942368951059055747527853, 15.42454877186249972215065254465, 16.87786511665621581619347554838, 17.88925107706472435939000862285, 20.29919635190717531301360978834, 22.22865695310673398848787743909, 23.54907924313031382848122917032, 25.58025207028805344157844624330, 26.52361721260073699095693304508, 28.1135403544478079613576312062, 29.30665311674081785383473822871, 32.08870033349041354025324709333, 33.24391057926943760889197511612, 33.90744720508931169689215969275, 35.75894705759697022805978721007, 37.04702419924149002034137809193, 38.55314542520657607957893503746, 40.28560516440286882351456756526, 41.66915456742049996686224142466, 43.237215036162091178319451506955, 44.551823474408698148274135952161

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