Properties

Degree 1
Conductor 59
Sign $-0.458 + 0.888i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.907 + 0.419i)2-s + (−0.947 − 0.319i)3-s + (0.647 − 0.762i)4-s + (−0.856 − 0.515i)5-s + (0.994 − 0.108i)6-s + (0.0541 − 0.998i)7-s + (−0.267 + 0.963i)8-s + (0.796 + 0.605i)9-s + (0.994 + 0.108i)10-s + (0.161 − 0.986i)11-s + (−0.856 + 0.515i)12-s + (−0.796 + 0.605i)13-s + (0.370 + 0.928i)14-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.0541 + 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.907 + 0.419i)2-s + (−0.947 − 0.319i)3-s + (0.647 − 0.762i)4-s + (−0.856 − 0.515i)5-s + (0.994 − 0.108i)6-s + (0.0541 − 0.998i)7-s + (−0.267 + 0.963i)8-s + (0.796 + 0.605i)9-s + (0.994 + 0.108i)10-s + (0.161 − 0.986i)11-s + (−0.856 + 0.515i)12-s + (−0.796 + 0.605i)13-s + (0.370 + 0.928i)14-s + (0.647 + 0.762i)15-s + (−0.161 − 0.986i)16-s + (0.0541 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.458 + 0.888i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.458 + 0.888i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $-0.458 + 0.888i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (11, \cdot )$
Sato-Tate  :  $\mu(58)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (1:\ ),\ -0.458 + 0.888i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08303106942 + 0.1363355763i$
$L(\frac12,\chi)$  $\approx$  $0.08303106942 + 0.1363355763i$
$L(\chi,1)$  $\approx$  0.3863772968 + 0.01252683903i
$L(1,\chi)$  $\approx$  0.3863772968 + 0.01252683903i

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

−31.98269000261680271714325724095, −30.65938993617036171081894442346, −29.68677269441781083859234514135, −28.404550439942444152721182715591, −27.72483441636692380663921508155, −26.94288969709111128944652875346, −25.59624379574595940626622622131, −24.26839165491530551217650016143, −22.68218887590127786844695290283, −21.970358459907088895928298740944, −20.55357526806739994425922492171, −19.255717004358478252587523431056, −18.1819026845602639553379201723, −17.33315118254130587670775496304, −15.8077503760827270000641390107, −15.19579410970777878974243417783, −12.31472460896595765369373156408, −11.86998869531146610214965335506, −10.57598288368840930862769090394, −9.45284013918275056371285092219, −7.769188157426725205092639169357, −6.517814430140836945364344101714, −4.53256056693696848064569538597, −2.59751564243716520936730598460, −0.14602617910164970444116118564, 1.24101200139244016074782577701, 4.35761372247536671681390682625, 6.05793852553743232729706158145, 7.322595215030541333212962059083, 8.38521557309953656276993336584, 10.231967945620713313342579296862, 11.23201365025866709644820673868, 12.405766600073577350422043586238, 14.2501274433540953275287705158, 15.99200355750121205237066879899, 16.68670618794806221739688607693, 17.53464370072614474117147107581, 19.11397028607259197197337951672, 19.68443239632213639180147399651, 21.37847312306133425730296441887, 23.212663269220148660204944901610, 23.86205655728338004393443168036, 24.659638503286899402205685777985, 26.48471321307950756929119222795, 27.24404679168582674161017754085, 28.194317951145875541594185045509, 29.27923030161414287236844611629, 30.13568864051915771542774783188, 32.0003501180154005449835080333, 33.12232347270739384970140594246

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