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Properties

Label 1-571-571.151-r1-0-0
Degree $1$
Conductor $571$
Sign $0.488 + 0.872i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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 + (−0.930 − 0.366i)3-s + (0.993 + 0.110i)4-s + (0.471 + 0.882i)5-s + (−0.909 − 0.416i)6-s + (−0.180 + 0.983i)7-s + (0.986 + 0.164i)8-s + (0.731 + 0.681i)9-s + (0.421 + 0.906i)10-s + (0.996 − 0.0880i)11-s + (−0.884 − 0.466i)12-s + (0.999 − 0.0220i)13-s + (−0.234 + 0.972i)14-s + (−0.115 − 0.993i)15-s + (0.975 + 0.218i)16-s + (0.660 + 0.750i)17-s + ⋯
L(s)  = 1  + (0.998 + 0.0550i)2-s + (−0.930 − 0.366i)3-s + (0.993 + 0.110i)4-s + (0.471 + 0.882i)5-s + (−0.909 − 0.416i)6-s + (−0.180 + 0.983i)7-s + (0.986 + 0.164i)8-s + (0.731 + 0.681i)9-s + (0.421 + 0.906i)10-s + (0.996 − 0.0880i)11-s + (−0.884 − 0.466i)12-s + (0.999 − 0.0220i)13-s + (−0.234 + 0.972i)14-s + (−0.115 − 0.993i)15-s + (0.975 + 0.218i)16-s + (0.660 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.473512921 + 2.035419145i\)
\(L(\frac12)\) \(\approx\) \(3.473512921 + 2.035419145i\)
\(L(1)\) \(\approx\) \(1.885807700 + 0.4748303414i\)
\(L(1)\) \(\approx\) \(1.885807700 + 0.4748303414i\)

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 \( 1 + (-0.930 - 0.366i)T \)
5 \( 1 + (0.471 + 0.882i)T \)
7 \( 1 + (-0.180 + 0.983i)T \)
11 \( 1 + (0.996 - 0.0880i)T \)
13 \( 1 + (0.999 - 0.0220i)T \)
17 \( 1 + (0.660 + 0.750i)T \)
19 \( 1 + (0.0605 - 0.998i)T \)
23 \( 1 + (-0.461 - 0.887i)T \)
29 \( 1 + (0.904 - 0.426i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (0.329 - 0.944i)T \)
41 \( 1 + (-0.137 + 0.990i)T \)
43 \( 1 + (0.874 + 0.485i)T \)
47 \( 1 + (0.298 + 0.954i)T \)
53 \( 1 + (-0.840 - 0.542i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.709 + 0.705i)T \)
67 \( 1 + (0.889 - 0.456i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (0.982 - 0.186i)T \)
79 \( 1 + (-0.583 + 0.812i)T \)
83 \( 1 + (0.815 - 0.578i)T \)
89 \( 1 + (-0.0935 - 0.995i)T \)
97 \( 1 + (-0.868 + 0.495i)T \)
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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

−22.99665344607639147261247804429, −22.13344740520654347018463654920, −21.34218045159720286605957090403, −20.53028635699462697342401244943, −20.123596975498006585841054651500, −18.75349184041501052267643960947, −17.48936216030355355793076181858, −16.7354601831043846937061262855, −16.32534397034737726688450487552, −15.4605434156896664875878186398, −14.09203822679269480258480273825, −13.664389538497065296769982542654, −12.53609489388848926186939185615, −11.97326320059683390978567079927, −11.04609727647884961767139700873, −10.14240752433744994040583206807, −9.36523039596709766366478978955, −7.79106455303580069706780173293, −6.67702606778896351668663627744, −5.93713844337166607710804648185, −5.14095716471473659516055399983, −4.10584904612711476577962313280, −3.59807827010500430543484253653, −1.560089654020850767489660058381, −0.89427123571519708413158962493, 1.29351112040415792273972645734, 2.328314381624323916977065455857, 3.409926952702898310575689299020, 4.5910672866210560622723346854, 5.81670232095741849652324465908, 6.20976427530770339890657594608, 6.85583328936770537361194680386, 8.123063512740240470722139689749, 9.55726642607403866066044772560, 10.78866791758049785716645736567, 11.22562913449852772410251197307, 12.22161906020142524426386515141, 12.81774716273926558699959593538, 13.88603185253254591014184396136, 14.58762578093055418811431568522, 15.58204041870282794085065634142, 16.28683372448080594303764841669, 17.32099488275486649941121982091, 18.127337825128431926575321677381, 18.98787289163450763966065764070, 19.76709541131921473500629159715, 21.37762004504844286700648588280, 21.59391453873521013477733902516, 22.46587158436847533442632970601, 22.945784278272877826810875360787

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