Properties

Degree 1
Conductor 59
Sign $0.985 + 0.167i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.976 − 0.214i)2-s + (−0.161 + 0.986i)3-s + (0.907 − 0.419i)4-s + (0.267 − 0.963i)5-s + (0.0541 + 0.998i)6-s + (−0.725 + 0.687i)7-s + (0.796 − 0.605i)8-s + (−0.947 − 0.319i)9-s + (0.0541 − 0.998i)10-s + (0.647 + 0.762i)11-s + (0.267 + 0.963i)12-s + (−0.947 + 0.319i)13-s + (−0.561 + 0.827i)14-s + (0.907 + 0.419i)15-s + (0.647 − 0.762i)16-s + (−0.725 − 0.687i)17-s + ⋯
L(s,χ)  = 1  + (0.976 − 0.214i)2-s + (−0.161 + 0.986i)3-s + (0.907 − 0.419i)4-s + (0.267 − 0.963i)5-s + (0.0541 + 0.998i)6-s + (−0.725 + 0.687i)7-s + (0.796 − 0.605i)8-s + (−0.947 − 0.319i)9-s + (0.0541 − 0.998i)10-s + (0.647 + 0.762i)11-s + (0.267 + 0.963i)12-s + (−0.947 + 0.319i)13-s + (−0.561 + 0.827i)14-s + (0.907 + 0.419i)15-s + (0.647 − 0.762i)16-s + (−0.725 − 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.985 + 0.167i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.985 + 0.167i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(59\)
\( \varepsilon \)  =  $0.985 + 0.167i$
motivic weight  =  \(0\)
character  :  $\chi_{59} (15, \cdot )$
Sato-Tate  :  $\mu(29)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 59,\ (0:\ ),\ 0.985 + 0.167i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.387607510 + 0.1167680686i$
$L(\frac12,\chi)$  $\approx$  $1.387607510 + 0.1167680686i$
$L(\chi,1)$  $\approx$  1.508463962 + 0.08481531945i
$L(1,\chi)$  $\approx$  1.508463962 + 0.08481531945i

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

−32.65334953679220798577364650220, −31.52464351400114408122071554817, −30.23342148230078782070215369958, −29.74199627959170273190860074728, −28.95153971034585600580190415902, −26.72776048904512612341070225019, −25.66349262772912103065066149343, −24.66705732510178914760211669079, −23.57631368737275733447677343544, −22.56678533913235240098720220747, −21.86914056426117998761431571316, −19.95648430988201749095248302825, −19.167292707675859844658095772905, −17.53498306832492685574833859721, −16.512471357343212266518081005831, −14.74677619970355568169924724491, −13.87844704542276412224476986626, −12.875590088787268005247660574361, −11.619491260162871851713883358866, −10.3600774054869020164908809358, −7.88213383905945127442032777019, −6.6725813019469667842420349125, −5.99449169139152971922503036983, −3.72813149356982443826337729882, −2.29753034135790387551959704672, 2.49109451445737076355742615845, 4.28841987423878623246870531953, 5.16890178052283248366819418478, 6.580625395109559139161686831592, 9.07225438145924413321123499541, 9.98423096324324110940261288890, 11.729667038570082809213932037769, 12.54226159868748305163625198041, 14.03847243413447223298613525216, 15.364607077958779910947830064065, 16.14989941810773491330388470394, 17.36162557129619110962369523444, 19.6955681900569964071840498940, 20.32082554257373960912473402921, 21.78970511515858584240988192677, 22.11444004735203991174442315584, 23.536712101472133143081230247944, 24.8488363254664635854397987709, 25.72841057496430257651164302098, 27.50927664780933635650327403155, 28.58010044737171183926664875021, 29.124194190024548037615036772832, 30.88239519115033518783571151060, 32.04755928208846124110429731098, 32.40167846677938688937057961454

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