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Properties

Label 1-59-59.21-r0-0-0
Degree $1$
Conductor $59$
Sign $0.418 + 0.908i$
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  + (0.468 + 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (0.976 + 0.214i)5-s + (0.267 − 0.963i)6-s + (0.796 + 0.605i)7-s + (−0.994 − 0.108i)8-s + (0.0541 + 0.998i)9-s + (0.267 + 0.963i)10-s + (−0.370 + 0.928i)11-s + (0.976 − 0.214i)12-s + (0.0541 − 0.998i)13-s + (−0.161 + 0.986i)14-s + (−0.561 − 0.827i)15-s + (−0.370 − 0.928i)16-s + (0.796 − 0.605i)17-s + ⋯
L(s)  = 1  + (0.468 + 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (0.976 + 0.214i)5-s + (0.267 − 0.963i)6-s + (0.796 + 0.605i)7-s + (−0.994 − 0.108i)8-s + (0.0541 + 0.998i)9-s + (0.267 + 0.963i)10-s + (−0.370 + 0.928i)11-s + (0.976 − 0.214i)12-s + (0.0541 − 0.998i)13-s + (−0.161 + 0.986i)14-s + (−0.561 − 0.827i)15-s + (−0.370 − 0.928i)16-s + (0.796 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.418 + 0.908i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.418 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8411635465 + 0.5388188788i\)
\(L(\frac12)\) \(\approx\) \(0.8411635465 + 0.5388188788i\)
\(L(1)\) \(\approx\) \(1.020844500 + 0.4420549957i\)
\(L(1)\) \(\approx\) \(1.020844500 + 0.4420549957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (0.468 + 0.883i)T \)
3 \( 1 + (-0.725 - 0.687i)T \)
5 \( 1 + (0.976 + 0.214i)T \)
7 \( 1 + (0.796 + 0.605i)T \)
11 \( 1 + (-0.370 + 0.928i)T \)
13 \( 1 + (0.0541 - 0.998i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (-0.947 - 0.319i)T \)
23 \( 1 + (-0.856 - 0.515i)T \)
29 \( 1 + (0.468 - 0.883i)T \)
31 \( 1 + (-0.947 + 0.319i)T \)
37 \( 1 + (-0.994 + 0.108i)T \)
41 \( 1 + (-0.856 + 0.515i)T \)
43 \( 1 + (-0.370 - 0.928i)T \)
47 \( 1 + (0.976 - 0.214i)T \)
53 \( 1 + (0.267 - 0.963i)T \)
61 \( 1 + (0.468 + 0.883i)T \)
67 \( 1 + (-0.994 - 0.108i)T \)
71 \( 1 + (0.976 - 0.214i)T \)
73 \( 1 + (-0.161 + 0.986i)T \)
79 \( 1 + (-0.725 + 0.687i)T \)
83 \( 1 + (0.647 + 0.762i)T \)
89 \( 1 + (0.468 - 0.883i)T \)
97 \( 1 + (-0.161 - 0.986i)T \)
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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

−32.57319150393906530549097432061, −31.52585523399322142073428166215, −29.89282013935989708940463925750, −29.3389771838051822516781516388, −28.223816394134513694072568464112, −27.32117065217976685031764204682, −26.106162325651886128067102323919, −24.045469109481315270338946450971, −23.47242512400384664852676394335, −21.77034054109840993993568506008, −21.42210683993985828717668798060, −20.461667357268039142105261589534, −18.729131662990142154745412502963, −17.52760833650774287448635453050, −16.48111024210723590169701972988, −14.63248067838380441945639912542, −13.72235008084928664325228404609, −12.24531372882803286227899919420, −10.94275338738938515643468377138, −10.2195435783581645590492723455, −8.84669549738490506884254294590, −6.14054223138762419796636835558, −5.09362674323528931763442234242, −3.793321643404110851774743083736, −1.586431888081172259316806824304, 2.29255163264460012232051563762, 4.9848030493549688935576834879, 5.78085827627988416066127355341, 7.09690126324663242313127367869, 8.35871372638492356506367060435, 10.25465127337739513965423806245, 12.0337429467743609818567040346, 12.99152039603517926615368624726, 14.189804236323544366895193486968, 15.38263828234717042202691007163, 16.96796017585186260367033455896, 17.83783475386755150713917507550, 18.4286165093624821294107352030, 20.803131440946918743237128042, 21.944847050470777235802603802980, 22.86856731507236973525418367974, 23.96691939207904381449768416277, 25.097267187436618685871525190552, 25.577871830633581509893712164456, 27.40877362205850327193574550562, 28.444386342078422837039263407818, 29.9913930766757972747481697979, 30.521507156391790966595694329574, 31.956811070903998133436522944883, 33.2575062746461354158508892063

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