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Properties

Label 1-571-571.31-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.860 + 0.509i$
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  + (0.945 + 0.324i)2-s + (0.546 + 0.837i)3-s + (0.789 + 0.614i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (−0.986 + 0.164i)7-s + (0.546 + 0.837i)8-s + (−0.401 + 0.915i)9-s + (−0.401 − 0.915i)10-s + (−0.401 + 0.915i)11-s + (−0.0825 + 0.996i)12-s + (−0.879 − 0.475i)13-s + (−0.986 − 0.164i)14-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (0.546 + 0.837i)3-s + (0.789 + 0.614i)4-s + (−0.677 − 0.735i)5-s + (0.245 + 0.969i)6-s + (−0.986 + 0.164i)7-s + (0.546 + 0.837i)8-s + (−0.401 + 0.915i)9-s + (−0.401 − 0.915i)10-s + (−0.401 + 0.915i)11-s + (−0.0825 + 0.996i)12-s + (−0.879 − 0.475i)13-s + (−0.986 − 0.164i)14-s + (0.245 − 0.969i)15-s + (0.245 + 0.969i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4890038077 + 1.785092043i\)
\(L(\frac12)\) \(\approx\) \(0.4890038077 + 1.785092043i\)
\(L(1)\) \(\approx\) \(1.254192877 + 0.9335341628i\)
\(L(1)\) \(\approx\) \(1.254192877 + 0.9335341628i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (0.546 + 0.837i)T \)
5 \( 1 + (-0.677 - 0.735i)T \)
7 \( 1 + (-0.986 + 0.164i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (-0.879 - 0.475i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (0.546 + 0.837i)T \)
23 \( 1 + (0.546 - 0.837i)T \)
29 \( 1 + (-0.879 - 0.475i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (-0.677 - 0.735i)T \)
43 \( 1 + (-0.401 - 0.915i)T \)
47 \( 1 + (0.245 + 0.969i)T \)
53 \( 1 + (0.945 - 0.324i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (0.945 + 0.324i)T \)
67 \( 1 + (0.945 - 0.324i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.879 + 0.475i)T \)
79 \( 1 + (0.789 - 0.614i)T \)
83 \( 1 + (0.245 + 0.969i)T \)
89 \( 1 + (0.245 + 0.969i)T \)
97 \( 1 + (0.789 + 0.614i)T \)
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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

−22.96077222050975184281137002555, −22.214681638826936714131011509790, −21.37418202021866256071699883823, −20.14388918579403232884674051605, −19.67786884388369335289019569681, −18.91207575618261481777617986413, −18.4493266857065102322697073226, −16.84106595496218844696202092474, −15.84533750362527421720533748921, −15.139223253295178984976641529738, −14.20328528485551344143871480590, −13.54465104521757187684199660168, −12.86621951176153692763091463315, −11.75825402133300824816583195368, −11.3633455156440692386184968663, −10.049916245351587932220051502201, −9.132288863220457758939310032209, −7.55087626591322526690941670842, −7.12281217483399287924444395795, −6.28376264248443044375704036781, −5.12743134193929793484494292571, −3.61364037271773511315519207364, −3.112458111897455317270093402864, −2.31275956189747212130887226062, −0.60260622745341604202879217385, 2.103842060293763956814463940049, 3.24377051317629842536942084243, 3.91013378734519138374425429303, 4.91311983685930686475492833371, 5.5506652158182462256146605051, 7.004746187601556920243066232029, 7.87296455664077745830049501414, 8.71240602463688059063699027093, 9.88971992879525561798100539445, 10.61716827880556841885574525583, 12.09699716892174516853608470071, 12.529649334418426213979934198549, 13.36941971472600652264749391231, 14.520544754091800800248531177831, 15.29689175983231967768177453919, 15.69457359875990110487328173402, 16.66238763366008399831522744071, 17.147248971624878887081585707930, 18.9908419582509180409871778595, 19.77990565168445527831197667948, 20.46165208263052903568689812415, 21.00377481947211104685181128266, 22.15382333774019874210718929166, 22.6561973231515120200055174644, 23.40092376603247002446301473575

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