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Properties

Label 1-59-59.11-r1-0-0
Degree $1$
Conductor $59$
Sign $-0.458 + 0.888i$
Analytic cond. $6.34043$
Root an. cond. $6.34043$
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  + (−0.907 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.458 + 0.888i$
Analytic conductor: \(6.34043\)
Root analytic conductor: \(6.34043\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (1:\ ),\ -0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08303106942 + 0.1363355763i\)
\(L(\frac12)\) \(\approx\) \(0.08303106942 + 0.1363355763i\)
\(L(1)\) \(\approx\) \(0.3863772968 + 0.01252683903i\)
\(L(1)\) \(\approx\) \(0.3863772968 + 0.01252683903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.907 + 0.419i)T \)
3 \( 1 + (-0.947 - 0.319i)T \)
5 \( 1 + (-0.856 - 0.515i)T \)
7 \( 1 + (0.0541 - 0.998i)T \)
11 \( 1 + (0.161 - 0.986i)T \)
13 \( 1 + (-0.796 + 0.605i)T \)
17 \( 1 + (0.0541 + 0.998i)T \)
19 \( 1 + (-0.725 + 0.687i)T \)
23 \( 1 + (-0.976 + 0.214i)T \)
29 \( 1 + (0.907 + 0.419i)T \)
31 \( 1 + (0.725 + 0.687i)T \)
37 \( 1 + (-0.267 - 0.963i)T \)
41 \( 1 + (0.976 + 0.214i)T \)
43 \( 1 + (0.161 + 0.986i)T \)
47 \( 1 + (0.856 - 0.515i)T \)
53 \( 1 + (-0.994 + 0.108i)T \)
61 \( 1 + (-0.907 + 0.419i)T \)
67 \( 1 + (-0.267 + 0.963i)T \)
71 \( 1 + (-0.856 + 0.515i)T \)
73 \( 1 + (0.370 + 0.928i)T \)
79 \( 1 + (-0.947 + 0.319i)T \)
83 \( 1 + (0.561 - 0.827i)T \)
89 \( 1 + (-0.907 - 0.419i)T \)
97 \( 1 + (0.370 - 0.928i)T \)
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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

−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