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Properties

Label 1-571-571.174-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: $\chi_{571} (174, \cdot )$
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

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

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