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Properties

Label 1-571-571.82-r0-0-0
Degree $1$
Conductor $571$
Sign $0.411 - 0.911i$
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  + (0.851 + 0.523i)2-s + (−0.821 − 0.569i)3-s + (0.451 + 0.892i)4-s + (−0.191 − 0.981i)5-s + (−0.401 − 0.915i)6-s + (0.245 − 0.969i)7-s + (−0.0825 + 0.996i)8-s + (0.350 + 0.936i)9-s + (0.350 − 0.936i)10-s + (0.635 − 0.771i)11-s + (0.137 − 0.990i)12-s + (0.975 − 0.218i)13-s + (0.716 − 0.697i)14-s + (−0.401 + 0.915i)15-s + (−0.592 + 0.805i)16-s + (−0.592 + 0.805i)17-s + ⋯
L(s)  = 1  + (0.851 + 0.523i)2-s + (−0.821 − 0.569i)3-s + (0.451 + 0.892i)4-s + (−0.191 − 0.981i)5-s + (−0.401 − 0.915i)6-s + (0.245 − 0.969i)7-s + (−0.0825 + 0.996i)8-s + (0.350 + 0.936i)9-s + (0.350 − 0.936i)10-s + (0.635 − 0.771i)11-s + (0.137 − 0.990i)12-s + (0.975 − 0.218i)13-s + (0.716 − 0.697i)14-s + (−0.401 + 0.915i)15-s + (−0.592 + 0.805i)16-s + (−0.592 + 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373932127 - 0.8874758753i\)
\(L(\frac12)\) \(\approx\) \(1.373932127 - 0.8874758753i\)
\(L(1)\) \(\approx\) \(1.302065047 - 0.2356503581i\)
\(L(1)\) \(\approx\) \(1.302065047 - 0.2356503581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.851 + 0.523i)T \)
3 \( 1 + (-0.821 - 0.569i)T \)
5 \( 1 + (-0.191 - 0.981i)T \)
7 \( 1 + (0.245 - 0.969i)T \)
11 \( 1 + (0.635 - 0.771i)T \)
13 \( 1 + (0.975 - 0.218i)T \)
17 \( 1 + (-0.592 + 0.805i)T \)
19 \( 1 + (-0.821 - 0.569i)T \)
23 \( 1 + (-0.0825 - 0.996i)T \)
29 \( 1 + (-0.298 + 0.954i)T \)
31 \( 1 + (-0.401 - 0.915i)T \)
37 \( 1 + (0.975 + 0.218i)T \)
41 \( 1 + (-0.191 - 0.981i)T \)
43 \( 1 + (0.350 - 0.936i)T \)
47 \( 1 + (0.993 + 0.110i)T \)
53 \( 1 + (0.851 - 0.523i)T \)
59 \( 1 + (0.245 - 0.969i)T \)
61 \( 1 + (0.0275 - 0.999i)T \)
67 \( 1 + (0.0275 + 0.999i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.298 - 0.954i)T \)
79 \( 1 + (-0.998 + 0.0550i)T \)
83 \( 1 + (0.993 + 0.110i)T \)
89 \( 1 + (-0.592 + 0.805i)T \)
97 \( 1 + (0.451 + 0.892i)T \)
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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

−22.94715118848611430424065352464, −22.727675819339375029562815121688, −21.64683644177310438015872501142, −21.38332522266784749193875675247, −20.295415791554843032382793058284, −19.27840260678174247101102292136, −18.32410735603094676240625834917, −17.85468666475795548533726804860, −16.41664169452605224681140656584, −15.467242086948821509106414963897, −15.078198175853633243374016462492, −14.23757053120639865432780176439, −13.070925699531140493486748043690, −11.96028091607937397069108429133, −11.54004639236398480302124271677, −10.83276818224284991761493434227, −9.87124511262467560529812359655, −9.04776381552578563978893018421, −7.275868016870959969477314642381, −6.274652314363508501984830657225, −5.79715719875353505886127848391, −4.53714969769257992762944784906, −3.85691709675900805549027078449, −2.71205410404541529394953684190, −1.56341158308508217262058306429, 0.73398714715899450732934097474, 1.970461802591598886966997237943, 3.84806001936720657171433993103, 4.35744992258290033905801589581, 5.47385107558504619605718022269, 6.26958325013096321636823447428, 7.064957259888293468629386207486, 8.16891190615948002075047893779, 8.76060058708576140668598093004, 10.74198548646075204767769799220, 11.186510828574485568136578617807, 12.21639483931347669576038716232, 13.11205066859327876803076243499, 13.412940261940859455491784041226, 14.503229146481996286594945340426, 15.71915034880876316658633735875, 16.51195002061323330917939723086, 17.00917213975816339669421461190, 17.63013516420962412613325884299, 18.92717651305966479064450009107, 20.02542994521604274562329222339, 20.64173184238159609807890628946, 21.727719367011051758806923536985, 22.36323613099498739599788502385, 23.49066437945334882836106659861

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