↑

Properties

Label 1-59-59.49-r0-0-0
Degree $1$
Conductor $59$
Sign $0.776 - 0.630i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.725 − 0.687i)2-s + (0.976 + 0.214i)3-s + (0.0541 + 0.998i)4-s + (−0.161 − 0.986i)5-s + (−0.561 − 0.827i)6-s + (0.468 + 0.883i)7-s + (0.647 − 0.762i)8-s + (0.907 + 0.419i)9-s + (−0.561 + 0.827i)10-s + (−0.994 − 0.108i)11-s + (−0.161 + 0.986i)12-s + (0.907 − 0.419i)13-s + (0.267 − 0.963i)14-s + (0.0541 − 0.998i)15-s + (−0.994 + 0.108i)16-s + (0.468 − 0.883i)17-s + ⋯
L(s)  = 1  + (−0.725 − 0.687i)2-s + (0.976 + 0.214i)3-s + (0.0541 + 0.998i)4-s + (−0.161 − 0.986i)5-s + (−0.561 − 0.827i)6-s + (0.468 + 0.883i)7-s + (0.647 − 0.762i)8-s + (0.907 + 0.419i)9-s + (−0.561 + 0.827i)10-s + (−0.994 − 0.108i)11-s + (−0.161 + 0.986i)12-s + (0.907 − 0.419i)13-s + (0.267 − 0.963i)14-s + (0.0541 − 0.998i)15-s + (−0.994 + 0.108i)16-s + (0.468 − 0.883i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(59\)
Sign: $0.776 - 0.630i$
Analytic conductor: \(0.273994\)
Root analytic conductor: \(0.273994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{59} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 59,\ (0:\ ),\ 0.776 - 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8048848979 - 0.2858023966i\)
\(L(\frac12)\) \(\approx\) \(0.8048848979 - 0.2858023966i\)
\(L(1)\) \(\approx\) \(0.9200618243 - 0.2475655236i\)
\(L(1)\) \(\approx\) \(0.9200618243 - 0.2475655236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.725 - 0.687i)T \)
3 \( 1 + (0.976 + 0.214i)T \)
5 \( 1 + (-0.161 - 0.986i)T \)
7 \( 1 + (0.468 + 0.883i)T \)
11 \( 1 + (-0.994 - 0.108i)T \)
13 \( 1 + (0.907 - 0.419i)T \)
17 \( 1 + (0.468 - 0.883i)T \)
19 \( 1 + (-0.856 - 0.515i)T \)
23 \( 1 + (-0.370 + 0.928i)T \)
29 \( 1 + (-0.725 + 0.687i)T \)
31 \( 1 + (-0.856 + 0.515i)T \)
37 \( 1 + (0.647 + 0.762i)T \)
41 \( 1 + (-0.370 - 0.928i)T \)
43 \( 1 + (-0.994 + 0.108i)T \)
47 \( 1 + (-0.161 + 0.986i)T \)
53 \( 1 + (-0.561 - 0.827i)T \)
61 \( 1 + (-0.725 - 0.687i)T \)
67 \( 1 + (0.647 - 0.762i)T \)
71 \( 1 + (-0.161 + 0.986i)T \)
73 \( 1 + (0.267 - 0.963i)T \)
79 \( 1 + (0.976 - 0.214i)T \)
83 \( 1 + (0.796 - 0.605i)T \)
89 \( 1 + (-0.725 + 0.687i)T \)
97 \( 1 + (0.267 + 0.963i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.12382884020675636865311864871, −31.82051009377291340309742269337, −30.579128703120813030136826342569, −29.70000268481977031826267460058, −28.0830592988398981012400576919, −26.6941025733322804749082078660, −26.272408674106758064589558305237, −25.36475244939546618343191968441, −23.8562553611674579341681760108, −23.25659821361456622879500579454, −21.15929016430377982742546930151, −20.02610322853433638470883393571, −18.78308177240669952578080202201, −18.21208235661886991053113876952, −16.61608377937411821057016338965, −15.17853525807624909859357830759, −14.45633250004032011703097373326, −13.32976585079756891721935333284, −10.89916683808331471572992594333, −10.05090252852671298617665116791, −8.31087222461070955955224562417, −7.57288711322194151013662373946, −6.31415889707735407447133021668, −3.98214649760118541914771008740, −1.97400827084787050957319562588, 1.79857945906799690698125381290, 3.3019209654954599518459723732, 5.00623149933112624259087106458, 7.78987959732711600543297504086, 8.579779170400324201111294385362, 9.51096710444605212451350374149, 11.059365532561473756376449918000, 12.543194290863139531287694969539, 13.46876944530064006097714314745, 15.41490150839856917115809860792, 16.27609902386079000430436594845, 17.98926353126827766608122761731, 18.91150382937242964074185992937, 20.21025783236082973190077773498, 20.89522383947509781569522778594, 21.72792702348551501918346845935, 23.835663151860165195309633762212, 25.2122537333744166221008576733, 25.79984179504967749323306504039, 27.39629174860116315936750170327, 27.86765061956764013052168672956, 29.11503996738093539423616490138, 30.516732025407514334915678041425, 31.44296275925523412604378088037, 32.11662562522289534142966462326

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