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Properties

Label 1-59-59.4-r0-0-0
Degree $1$
Conductor $59$
Sign $0.985 - 0.167i$
Analytic cond. $0.273994$
Root an. cond. $0.273994$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387607510 - 0.1167680686i\)
\(L(\frac12)\) \(\approx\) \(1.387607510 - 0.1167680686i\)
\(L(1)\) \(\approx\) \(1.508463962 - 0.08481531945i\)
\(L(1)\) \(\approx\) \(1.508463962 - 0.08481531945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.976 + 0.214i)T \)
3 \( 1 + (-0.161 - 0.986i)T \)
5 \( 1 + (0.267 + 0.963i)T \)
7 \( 1 + (-0.725 - 0.687i)T \)
11 \( 1 + (0.647 - 0.762i)T \)
13 \( 1 + (-0.947 - 0.319i)T \)
17 \( 1 + (-0.725 + 0.687i)T \)
19 \( 1 + (-0.370 + 0.928i)T \)
23 \( 1 + (-0.994 - 0.108i)T \)
29 \( 1 + (0.976 - 0.214i)T \)
31 \( 1 + (-0.370 - 0.928i)T \)
37 \( 1 + (0.796 - 0.605i)T \)
41 \( 1 + (-0.994 + 0.108i)T \)
43 \( 1 + (0.647 + 0.762i)T \)
47 \( 1 + (0.267 - 0.963i)T \)
53 \( 1 + (0.0541 - 0.998i)T \)
61 \( 1 + (0.976 + 0.214i)T \)
67 \( 1 + (0.796 + 0.605i)T \)
71 \( 1 + (0.267 - 0.963i)T \)
73 \( 1 + (-0.561 - 0.827i)T \)
79 \( 1 + (-0.161 + 0.986i)T \)
83 \( 1 + (0.468 - 0.883i)T \)
89 \( 1 + (0.976 - 0.214i)T \)
97 \( 1 + (-0.561 + 0.827i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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