↑

Properties

Label 1-571-571.270-r0-0-0
Degree $1$
Conductor $571$
Sign $0.963 + 0.266i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.962 + 0.272i)2-s + (−0.298 − 0.954i)3-s + (0.851 − 0.523i)4-s + (0.635 + 0.771i)5-s + (0.546 + 0.837i)6-s + (0.789 + 0.614i)7-s + (−0.677 + 0.735i)8-s + (−0.821 + 0.569i)9-s + (−0.821 − 0.569i)10-s + (0.904 + 0.426i)11-s + (−0.754 − 0.656i)12-s + (0.993 + 0.110i)13-s + (−0.926 − 0.376i)14-s + (0.546 − 0.837i)15-s + (0.451 − 0.892i)16-s + (0.451 − 0.892i)17-s + ⋯
L(s)  = 1  + (−0.962 + 0.272i)2-s + (−0.298 − 0.954i)3-s + (0.851 − 0.523i)4-s + (0.635 + 0.771i)5-s + (0.546 + 0.837i)6-s + (0.789 + 0.614i)7-s + (−0.677 + 0.735i)8-s + (−0.821 + 0.569i)9-s + (−0.821 − 0.569i)10-s + (0.904 + 0.426i)11-s + (−0.754 − 0.656i)12-s + (0.993 + 0.110i)13-s + (−0.926 − 0.376i)14-s + (0.546 − 0.837i)15-s + (0.451 − 0.892i)16-s + (0.451 − 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.963 + 0.266i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.963 + 0.266i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064788088 + 0.1446095583i\)
\(L(\frac12)\) \(\approx\) \(1.064788088 + 0.1446095583i\)
\(L(1)\) \(\approx\) \(0.8515006411 + 0.03201517473i\)
\(L(1)\) \(\approx\) \(0.8515006411 + 0.03201517473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.962 + 0.272i)T \)
3 \( 1 + (-0.298 - 0.954i)T \)
5 \( 1 + (0.635 + 0.771i)T \)
7 \( 1 + (0.789 + 0.614i)T \)
11 \( 1 + (0.904 + 0.426i)T \)
13 \( 1 + (0.993 + 0.110i)T \)
17 \( 1 + (0.451 - 0.892i)T \)
19 \( 1 + (-0.298 - 0.954i)T \)
23 \( 1 + (-0.677 - 0.735i)T \)
29 \( 1 + (-0.592 + 0.805i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.993 - 0.110i)T \)
41 \( 1 + (0.635 + 0.771i)T \)
43 \( 1 + (-0.821 - 0.569i)T \)
47 \( 1 + (-0.998 + 0.0550i)T \)
53 \( 1 + (-0.962 - 0.272i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (0.716 + 0.697i)T \)
67 \( 1 + (0.716 - 0.697i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.592 - 0.805i)T \)
79 \( 1 + (0.0275 - 0.999i)T \)
83 \( 1 + (-0.998 + 0.0550i)T \)
89 \( 1 + (0.451 - 0.892i)T \)
97 \( 1 + (0.851 - 0.523i)T \)
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

−23.2641341777381547902591957456, −21.97744978257108079041581922562, −21.26936170258493512938866804444, −20.72838629714660244578796382851, −20.138492224866685249077069072485, −19.09065135615180082815408620386, −17.88982056168096761006844562935, −17.22779504506575167247597923803, −16.73792468925363963907671278564, −16.04235839092207054092830079616, −14.923242391817126392112621393863, −14.00495544461854494063363069418, −12.827433133672399745445852529648, −11.63648722082811143951614880951, −11.15379166634707032551195806946, −10.074107219933591306553466247165, −9.59847256945669554782311793365, −8.46043858322835247741727687519, −8.05225523165810773998665191708, −6.27242217949081382105140167722, −5.74194594767293934476516323186, −4.2025238700370764797260755283, −3.62436688765154686475751321351, −1.84544255988415791402077725634, −0.97353554548750697513717621445, 1.20522048406864431886137834123, 1.99910747867487015522360620256, 2.926656213717192911448361688291, 5.024606553982481085075918754906, 6.08410404204305470062584452508, 6.64313537395290685479464179057, 7.4854315736361940157206363338, 8.513286596185184000784864321687, 9.24431197652487557875748684040, 10.4233198436817662216508876891, 11.38921526934084298053530653804, 11.74039903482674318915781093865, 13.07897923855003091586509657910, 14.34836740911268817310781935300, 14.58064914435878589063474726419, 15.90015662795383993486524191160, 16.88812546328274153418005622858, 17.79977037204206230177508793229, 18.13337433004406403524831734820, 18.740600451981146352050384915769, 19.69061604145122105284770744337, 20.58928167441369634782123873949, 21.61163072525105265575531578453, 22.57877693757286170001715609899, 23.493911763134385659533063578363

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