Properties

Label 1-571-571.376-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.905 + 0.424i$
Analytic cond. $2.65171$
Root an. cond. $2.65171$
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  + (−0.998 − 0.0550i)2-s + (0.635 − 0.771i)3-s + (0.993 + 0.110i)4-s + (0.137 − 0.990i)5-s + (−0.677 + 0.735i)6-s + (−0.879 − 0.475i)7-s + (−0.986 − 0.164i)8-s + (−0.191 − 0.981i)9-s + (−0.191 + 0.981i)10-s + (−0.754 + 0.656i)11-s + (0.716 − 0.697i)12-s + (−0.821 − 0.569i)13-s + (0.851 + 0.523i)14-s + (−0.677 − 0.735i)15-s + (0.975 + 0.218i)16-s + (0.975 + 0.218i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0550i)2-s + (0.635 − 0.771i)3-s + (0.993 + 0.110i)4-s + (0.137 − 0.990i)5-s + (−0.677 + 0.735i)6-s + (−0.879 − 0.475i)7-s + (−0.986 − 0.164i)8-s + (−0.191 − 0.981i)9-s + (−0.191 + 0.981i)10-s + (−0.754 + 0.656i)11-s + (0.716 − 0.697i)12-s + (−0.821 − 0.569i)13-s + (0.851 + 0.523i)14-s + (−0.677 − 0.735i)15-s + (0.975 + 0.218i)16-s + (0.975 + 0.218i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1089096118 - 0.4889061033i\)
\(L(\frac12)\) \(\approx\) \(-0.1089096118 - 0.4889061033i\)
\(L(1)\) \(\approx\) \(0.5132516793 - 0.3859190461i\)
\(L(1)\) \(\approx\) \(0.5132516793 - 0.3859190461i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.998 - 0.0550i)T \)
3 \( 1 + (0.635 - 0.771i)T \)
5 \( 1 + (0.137 - 0.990i)T \)
7 \( 1 + (-0.879 - 0.475i)T \)
11 \( 1 + (-0.754 + 0.656i)T \)
13 \( 1 + (-0.821 - 0.569i)T \)
17 \( 1 + (0.975 + 0.218i)T \)
19 \( 1 + (0.635 - 0.771i)T \)
23 \( 1 + (-0.986 + 0.164i)T \)
29 \( 1 + (0.904 - 0.426i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.821 + 0.569i)T \)
41 \( 1 + (0.137 - 0.990i)T \)
43 \( 1 + (-0.191 + 0.981i)T \)
47 \( 1 + (-0.298 - 0.954i)T \)
53 \( 1 + (-0.998 + 0.0550i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (0.451 + 0.892i)T \)
67 \( 1 + (0.451 - 0.892i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.904 + 0.426i)T \)
79 \( 1 + (-0.592 - 0.805i)T \)
83 \( 1 + (-0.298 - 0.954i)T \)
89 \( 1 + (0.975 + 0.218i)T \)
97 \( 1 + (0.993 + 0.110i)T \)
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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

−23.89658581009431235712581482954, −22.665027090773184955091546399336, −21.8184841096030426601920857482, −21.250761660979907391245740957744, −20.24636130310633588339453765173, −19.31972412443688808005437964170, −18.8381968931383516254617917143, −18.159870867950038803328886415098, −16.8158671821852513447494567343, −16.12252035868170499427252853237, −15.56707894518979715696869301605, −14.52096060170947264803316698895, −13.98366356806335894579004208385, −12.452065062308884521725268699353, −11.45632415342809172248189531026, −10.41103517842298863104196195987, −9.913563901645704507966770435511, −9.260790881240403897322253499478, −8.09211493778963721436587552255, −7.42169161382131281194044662685, −6.239568763656445189691117865651, −5.38776886639449836073493726809, −3.53490676674906874087209287985, −2.91233337148008608416205279521, −2.062784861445717570071854200833, 0.31205549979357793642550527503, 1.45347843946589694021309631624, 2.553920584490506276540115096275, 3.4931824605410972661607053591, 5.16694613452998847621552651523, 6.32482125628719178395523833778, 7.394774153874334137973564794780, 7.86392660398239099601294549487, 8.865626488791207614933843793819, 9.78372540279082335707299528073, 10.21393055771374088824775786053, 11.93336840578235059451577139636, 12.47938204258279236567374313480, 13.17512937141178879635564547037, 14.25463201420510204410869888768, 15.50375239896709600668025089483, 16.0794715723963393483830163216, 17.21203815646848630254894728690, 17.688708650856983889360768259829, 18.65664347327203652918040623395, 19.64314644825019241093860704986, 19.99069185067270133375757547913, 20.64962905854256198135757240756, 21.62284421528596557984213986819, 23.09847067138018431706905409358

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