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Properties

Label 1-59-59.35-r0-0-0
Degree $1$
Conductor $59$
Sign $-0.999 + 0.0310i$
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  + (−0.856 + 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (0.796 + 0.605i)6-s + (−0.947 + 0.319i)7-s + (0.0541 + 0.998i)8-s + (−0.725 + 0.687i)9-s + (0.796 − 0.605i)10-s + (−0.561 + 0.827i)11-s + (−0.994 − 0.108i)12-s + (−0.725 − 0.687i)13-s + (0.647 − 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯
L(s)  = 1  + (−0.856 + 0.515i)2-s + (−0.370 − 0.928i)3-s + (0.468 − 0.883i)4-s + (−0.994 + 0.108i)5-s + (0.796 + 0.605i)6-s + (−0.947 + 0.319i)7-s + (0.0541 + 0.998i)8-s + (−0.725 + 0.687i)9-s + (0.796 − 0.605i)10-s + (−0.561 + 0.827i)11-s + (−0.994 − 0.108i)12-s + (−0.725 − 0.687i)13-s + (0.647 − 0.762i)14-s + (0.468 + 0.883i)15-s + (−0.561 − 0.827i)16-s + (−0.947 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.999 + 0.0310i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ -0.999 + 0.0310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0004992210914 - 0.03215327025i\)
\(L(\frac12)\) \(\approx\) \(0.0004992210914 - 0.03215327025i\)
\(L(1)\) \(\approx\) \(0.3316177767 + 0.02308879173i\)
\(L(1)\) \(\approx\) \(0.3316177767 + 0.02308879173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.856 + 0.515i)T \)
3 \( 1 + (-0.370 - 0.928i)T \)
5 \( 1 + (-0.994 + 0.108i)T \)
7 \( 1 + (-0.947 + 0.319i)T \)
11 \( 1 + (-0.561 + 0.827i)T \)
13 \( 1 + (-0.725 - 0.687i)T \)
17 \( 1 + (-0.947 - 0.319i)T \)
19 \( 1 + (-0.161 - 0.986i)T \)
23 \( 1 + (0.267 + 0.963i)T \)
29 \( 1 + (-0.856 - 0.515i)T \)
31 \( 1 + (-0.161 + 0.986i)T \)
37 \( 1 + (0.0541 - 0.998i)T \)
41 \( 1 + (0.267 - 0.963i)T \)
43 \( 1 + (-0.561 - 0.827i)T \)
47 \( 1 + (-0.994 - 0.108i)T \)
53 \( 1 + (0.796 + 0.605i)T \)
61 \( 1 + (-0.856 + 0.515i)T \)
67 \( 1 + (0.0541 + 0.998i)T \)
71 \( 1 + (-0.994 - 0.108i)T \)
73 \( 1 + (0.647 - 0.762i)T \)
79 \( 1 + (-0.370 + 0.928i)T \)
83 \( 1 + (0.907 - 0.419i)T \)
89 \( 1 + (-0.856 - 0.515i)T \)
97 \( 1 + (0.647 + 0.762i)T \)
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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

−33.49152363861265135618303866075, −31.981055114290506390008476892071, −31.19866926233840896484328366610, −29.42876910059118630200510909496, −28.76575697067933057131213170914, −27.62865295484397132012909263060, −26.62113082891310420429265920693, −26.2097271346278434635932833901, −24.281464132835949995142262847954, −22.81596698281219581909110657636, −21.82040561197918729360563918406, −20.55800516385480493679141078490, −19.5976686104267744831184297689, −18.564520954106162122583012682275, −16.66643822501373817458042447846, −16.40956571337656722606627174044, −15.08837832658481974532451076598, −12.85841983283730429581749808814, −11.58114111818600009927695026520, −10.628552681589691909430402689022, −9.4480796803001560987847359329, −8.20575634992234592380777762500, −6.55681694493927345072790421214, −4.28612756241541179007995266592, −3.11991320000170495507024578230, 0.05165411900596720614647235625, 2.51255355890100796187765340625, 5.27159898837186661074764008919, 6.90083495263581498235577154970, 7.535489335422812287671596555126, 9.02359088925234431837890783204, 10.647515045757100436900804867704, 11.931120675425586191549831427386, 13.151568482007144488829088461186, 15.09920727874262454036763364669, 15.9213453393103502767474443731, 17.34292438507883417701231077721, 18.29299098317296604304553744857, 19.47463435084816700440323766366, 19.94533788790444590001107100698, 22.51060498671244058868639125193, 23.33638391757830074836446683343, 24.38776401594500979048063184350, 25.43089314429587255688514738975, 26.48094312247498375689976105946, 27.8434355113590320971662219015, 28.65501129256885376083679385250, 29.679249638215407981636338173788, 31.05364832045977246869775732830, 32.192202023681935166576706264481

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