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Properties

Label 1-571-571.233-r0-0-0
Degree $1$
Conductor $571$
Sign $0.940 - 0.340i$
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  + (−0.191 − 0.981i)2-s + (−0.360 − 0.932i)3-s + (−0.926 + 0.376i)4-s + (−0.319 − 0.947i)5-s + (−0.846 + 0.533i)6-s + (−0.461 − 0.887i)7-s + (0.546 + 0.837i)8-s + (−0.739 + 0.673i)9-s + (−0.868 + 0.495i)10-s + (0.00551 + 0.999i)11-s + (0.685 + 0.728i)12-s + (−0.381 + 0.924i)13-s + (−0.782 + 0.622i)14-s + (−0.768 + 0.639i)15-s + (0.716 − 0.697i)16-s + (−0.989 + 0.142i)17-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (−0.360 − 0.932i)3-s + (−0.926 + 0.376i)4-s + (−0.319 − 0.947i)5-s + (−0.846 + 0.533i)6-s + (−0.461 − 0.887i)7-s + (0.546 + 0.837i)8-s + (−0.739 + 0.673i)9-s + (−0.868 + 0.495i)10-s + (0.00551 + 0.999i)11-s + (0.685 + 0.728i)12-s + (−0.381 + 0.924i)13-s + (−0.782 + 0.622i)14-s + (−0.768 + 0.639i)15-s + (0.716 − 0.697i)16-s + (−0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.940 - 0.340i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.940 - 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3715707321 - 0.06520319352i\)
\(L(\frac12)\) \(\approx\) \(0.3715707321 - 0.06520319352i\)
\(L(1)\) \(\approx\) \(0.4322912217 - 0.3884957258i\)
\(L(1)\) \(\approx\) \(0.4322912217 - 0.3884957258i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (-0.360 - 0.932i)T \)
5 \( 1 + (-0.319 - 0.947i)T \)
7 \( 1 + (-0.461 - 0.887i)T \)
11 \( 1 + (0.00551 + 0.999i)T \)
13 \( 1 + (-0.381 + 0.924i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.775 - 0.631i)T \)
23 \( 1 + (0.965 + 0.261i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (-0.997 + 0.0770i)T \)
41 \( 1 + (-0.298 + 0.954i)T \)
43 \( 1 + (0.411 - 0.911i)T \)
47 \( 1 + (-0.962 - 0.272i)T \)
53 \( 1 + (0.731 - 0.681i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (-0.857 - 0.514i)T \)
67 \( 1 + (0.224 + 0.974i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.565 + 0.824i)T \)
79 \( 1 + (-0.899 + 0.436i)T \)
83 \( 1 + (-0.0385 - 0.999i)T \)
89 \( 1 + (0.884 + 0.466i)T \)
97 \( 1 + (0.528 - 0.849i)T \)
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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

−22.86221023312060722514826955026, −22.663340396543153653471310660780, −22.02682517696475284505116725767, −21.114446277871662242594525551460, −19.72211243321231832436835059729, −18.90916781132979046414426817322, −18.16912200612075552399330660810, −17.32492807516698910875441475033, −16.39203243631623618464953536052, −15.49252759038785832826201776677, −15.30842568249331480096292161432, −14.34348994364586196098128183732, −13.37699974233181582651524674477, −12.08086983442539472533964261464, −11.07810252313509551279094078018, −10.25877050457363083502883968593, −9.39405985800028968266704312122, −8.56696206778956182011452573373, −7.557087060477313491249709301969, −6.30536085534240537783968741319, −5.85052207882066022081058721526, −4.84239990021092909785496622894, −3.610116302170787730308254672402, −2.81631999854315404568888809354, −0.25569768612745246489891721944, 1.10777807423058087081886739243, 1.90185306777857551012500203287, 3.25729229343902859665100041698, 4.5539844013230257647474887265, 5.06100460303395569225914737664, 6.858955292348818862278359756678, 7.381703875753016190262058011737, 8.63718535330418415242023150289, 9.327219055915292614201942381366, 10.43253926874096650220532640141, 11.43135794092458183143968138004, 12.09559576477852468648039420359, 12.928752202593151227650485510814, 13.38905870137602781680997283686, 14.32585879904661998124356091037, 15.9036313734067410015210223733, 16.94160258756865647687675944338, 17.367262659764725741022029976888, 18.22878922024276451008008135213, 19.3540950570259504010363860813, 19.83710703740578501066591828761, 20.32928546258948536387566462225, 21.45645667204816627087457265232, 22.48398318358368631867546799914, 23.19309449339393495096155133066

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