Properties

Label 1-59-59.57-r0-0-0
Degree $1$
Conductor $59$
Sign $0.946 - 0.322i$
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  + (−0.994 − 0.108i)2-s + (0.647 − 0.762i)3-s + (0.976 + 0.214i)4-s + (0.796 + 0.605i)5-s + (−0.725 + 0.687i)6-s + (−0.370 + 0.928i)7-s + (−0.947 − 0.319i)8-s + (−0.161 − 0.986i)9-s + (−0.725 − 0.687i)10-s + (0.907 − 0.419i)11-s + (0.796 − 0.605i)12-s + (−0.161 + 0.986i)13-s + (0.468 − 0.883i)14-s + (0.976 − 0.214i)15-s + (0.907 + 0.419i)16-s + (−0.370 − 0.928i)17-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s + (0.647 − 0.762i)3-s + (0.976 + 0.214i)4-s + (0.796 + 0.605i)5-s + (−0.725 + 0.687i)6-s + (−0.370 + 0.928i)7-s + (−0.947 − 0.319i)8-s + (−0.161 − 0.986i)9-s + (−0.725 − 0.687i)10-s + (0.907 − 0.419i)11-s + (0.796 − 0.605i)12-s + (−0.161 + 0.986i)13-s + (0.468 − 0.883i)14-s + (0.976 − 0.214i)15-s + (0.907 + 0.419i)16-s + (−0.370 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.946 - 0.322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7729553751 - 0.1281081732i\)
\(L(\frac12)\) \(\approx\) \(0.7729553751 - 0.1281081732i\)
\(L(1)\) \(\approx\) \(0.8699722112 - 0.1150091607i\)
\(L(1)\) \(\approx\) \(0.8699722112 - 0.1150091607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.994 - 0.108i)T \)
3 \( 1 + (0.647 - 0.762i)T \)
5 \( 1 + (0.796 + 0.605i)T \)
7 \( 1 + (-0.370 + 0.928i)T \)
11 \( 1 + (0.907 - 0.419i)T \)
13 \( 1 + (-0.161 + 0.986i)T \)
17 \( 1 + (-0.370 - 0.928i)T \)
19 \( 1 + (-0.561 - 0.827i)T \)
23 \( 1 + (0.0541 - 0.998i)T \)
29 \( 1 + (-0.994 + 0.108i)T \)
31 \( 1 + (-0.561 + 0.827i)T \)
37 \( 1 + (-0.947 + 0.319i)T \)
41 \( 1 + (0.0541 + 0.998i)T \)
43 \( 1 + (0.907 + 0.419i)T \)
47 \( 1 + (0.796 - 0.605i)T \)
53 \( 1 + (-0.725 + 0.687i)T \)
61 \( 1 + (-0.994 - 0.108i)T \)
67 \( 1 + (-0.947 - 0.319i)T \)
71 \( 1 + (0.796 - 0.605i)T \)
73 \( 1 + (0.468 - 0.883i)T \)
79 \( 1 + (0.647 + 0.762i)T \)
83 \( 1 + (-0.856 + 0.515i)T \)
89 \( 1 + (-0.994 + 0.108i)T \)
97 \( 1 + (0.468 + 0.883i)T \)
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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

−33.02804759350682356826985250899, −32.106199169998208053323858326501, −30.23984408333760028519004147850, −29.34839660115503782234392709926, −27.95296797432729974510238841711, −27.29678083614187768200834701801, −25.93257714700422025424889225196, −25.42122314269954920922395579649, −24.23842504249924867672569954634, −22.36689442206116285919769843649, −20.930514956157051614599277156555, −20.18998732887755653765005288006, −19.30645362555402546132424187142, −17.3500447425336365037194287494, −16.89003047574517967417962551146, −15.52097620805014956497796900625, −14.299568245640532903947353157871, −12.79872063619922615121086303879, −10.72381404348592768851307916853, −9.8519465276090999656422278163, −8.92200555972689558086781730932, −7.56613229027334630687019691291, −5.81455223519433373231853438983, −3.81333114379183410466339962635, −1.823795071437304518009333758386, 1.89646578021428421685707350726, 2.95389588966292163140399241853, 6.27407014695522559153281407065, 7.00219085444941938364359698458, 8.891507428287931941294972229529, 9.35131422118551947583482639402, 11.20938427841857891161021170661, 12.43758956754971704379066586348, 13.99538966567054412755799712147, 15.13867404790896919380082166965, 16.76400467775849880394240972778, 18.070377623672261359519594098749, 18.78385723946253465055466637442, 19.67002198795333874814669307694, 21.101561019533984629572037245109, 22.191562671403397361386383650277, 24.27756549942222633629308680656, 25.05209429973867646367142770013, 25.90189291038906939773200351497, 26.80293934136939135433809241183, 28.41447501039063468168318398490, 29.34287395614387229120258235162, 30.13923589104200638701243122309, 31.30056170628975503716787784250, 32.76578865919736527629975583816

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