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Properties

Label 1-59-59.41-r0-0-0
Degree $1$
Conductor $59$
Sign $-0.503 + 0.864i$
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  + (0.0541 + 0.998i)2-s + (0.907 + 0.419i)3-s + (−0.994 + 0.108i)4-s + (−0.947 + 0.319i)5-s + (−0.370 + 0.928i)6-s + (−0.561 + 0.827i)7-s + (−0.161 − 0.986i)8-s + (0.647 + 0.762i)9-s + (−0.370 − 0.928i)10-s + (0.976 + 0.214i)11-s + (−0.947 − 0.319i)12-s + (0.647 − 0.762i)13-s + (−0.856 − 0.515i)14-s + (−0.994 − 0.108i)15-s + (0.976 − 0.214i)16-s + (−0.561 − 0.827i)17-s + ⋯
L(s)  = 1  + (0.0541 + 0.998i)2-s + (0.907 + 0.419i)3-s + (−0.994 + 0.108i)4-s + (−0.947 + 0.319i)5-s + (−0.370 + 0.928i)6-s + (−0.561 + 0.827i)7-s + (−0.161 − 0.986i)8-s + (0.647 + 0.762i)9-s + (−0.370 − 0.928i)10-s + (0.976 + 0.214i)11-s + (−0.947 − 0.319i)12-s + (0.647 − 0.762i)13-s + (−0.856 − 0.515i)14-s + (−0.994 − 0.108i)15-s + (0.976 − 0.214i)16-s + (−0.561 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $-0.503 + 0.864i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ -0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4724651497 + 0.8218949925i\)
\(L(\frac12)\) \(\approx\) \(0.4724651497 + 0.8218949925i\)
\(L(1)\) \(\approx\) \(0.7913592201 + 0.7140664905i\)
\(L(1)\) \(\approx\) \(0.7913592201 + 0.7140664905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.0541 + 0.998i)T \)
3 \( 1 + (0.907 + 0.419i)T \)
5 \( 1 + (-0.947 + 0.319i)T \)
7 \( 1 + (-0.561 + 0.827i)T \)
11 \( 1 + (0.976 + 0.214i)T \)
13 \( 1 + (0.647 - 0.762i)T \)
17 \( 1 + (-0.561 - 0.827i)T \)
19 \( 1 + (0.468 + 0.883i)T \)
23 \( 1 + (-0.725 - 0.687i)T \)
29 \( 1 + (0.0541 - 0.998i)T \)
31 \( 1 + (0.468 - 0.883i)T \)
37 \( 1 + (-0.161 + 0.986i)T \)
41 \( 1 + (-0.725 + 0.687i)T \)
43 \( 1 + (0.976 - 0.214i)T \)
47 \( 1 + (-0.947 - 0.319i)T \)
53 \( 1 + (-0.370 + 0.928i)T \)
61 \( 1 + (0.0541 + 0.998i)T \)
67 \( 1 + (-0.161 - 0.986i)T \)
71 \( 1 + (-0.947 - 0.319i)T \)
73 \( 1 + (-0.856 - 0.515i)T \)
79 \( 1 + (0.907 - 0.419i)T \)
83 \( 1 + (0.267 - 0.963i)T \)
89 \( 1 + (0.0541 - 0.998i)T \)
97 \( 1 + (-0.856 + 0.515i)T \)
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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

−32.16175028494414226806947074010, −30.9285976442369004329097706619, −30.39078063554425798008104431230, −29.22589485603228113082446163020, −27.949111332278771348353036442297, −26.78243767743268039412681419746, −26.00781026805464270800807122163, −24.14524942131209979509800778839, −23.398233308480412684915236445884, −21.96081614738663968607129604568, −20.55851272363667248776018134159, −19.58784185333092349948344770365, −19.31289107155858975199334057272, −17.66966060774092317681976286262, −16.00307443192811636765131360230, −14.37016221924225736826201037786, −13.4063878237202883113137935621, −12.294732130621471092176061881379, −11.05514429434442555874161573499, −9.38310704988558958879584716718, −8.44820013396335680084874971848, −6.88375359472276414871814547490, −4.165780931930198278425220404737, −3.43489143166072456936885112972, −1.35565643022649202433073828558, 3.19347431815291650069126697775, 4.38408342277545014569771863306, 6.2769526201079289552213282744, 7.75768342321859304183376174351, 8.73775112273358002538491649760, 9.92091900432783097607164880191, 12.02234709432445546678810831544, 13.51573956194093463510111820746, 14.787724606982527910309516563745, 15.54014852120526206598637839783, 16.37355747596174277693270533399, 18.29662360239919148799836361733, 19.19026411289075225459438732132, 20.45024153952669322211341247904, 22.24899327661410316743238534441, 22.73183480393438186848957355353, 24.499416618853437886369135359994, 25.21220617059275759774493298717, 26.2877613142724194400475060697, 27.26367040141587109691613414468, 28.0500989930105171191533421190, 30.33763368303125551534863023840, 31.23136108525366103700277784888, 32.032142814258550313870196285767, 32.93108572131548626367492408750

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