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Properties

Label 1-571-571.11-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.791 + 0.610i$
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  + (0.0275 + 0.999i)2-s + (−0.982 + 0.186i)3-s + (−0.998 + 0.0550i)4-s + (−0.857 + 0.514i)5-s + (−0.213 − 0.976i)6-s + (−0.768 − 0.639i)7-s + (−0.0825 − 0.996i)8-s + (0.930 − 0.366i)9-s + (−0.537 − 0.843i)10-s + (0.999 + 0.0440i)11-s + (0.970 − 0.240i)12-s + (−0.999 − 0.0110i)13-s + (0.618 − 0.785i)14-s + (0.746 − 0.665i)15-s + (0.993 − 0.110i)16-s + (0.411 + 0.911i)17-s + ⋯
L(s)  = 1  + (0.0275 + 0.999i)2-s + (−0.982 + 0.186i)3-s + (−0.998 + 0.0550i)4-s + (−0.857 + 0.514i)5-s + (−0.213 − 0.976i)6-s + (−0.768 − 0.639i)7-s + (−0.0825 − 0.996i)8-s + (0.930 − 0.366i)9-s + (−0.537 − 0.843i)10-s + (0.999 + 0.0440i)11-s + (0.970 − 0.240i)12-s + (−0.999 − 0.0110i)13-s + (0.618 − 0.785i)14-s + (0.746 − 0.665i)15-s + (0.993 − 0.110i)16-s + (0.411 + 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.791 + 0.610i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.791 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1606453118 + 0.4712001933i\)
\(L(\frac12)\) \(\approx\) \(0.1606453118 + 0.4712001933i\)
\(L(1)\) \(\approx\) \(0.4622962614 + 0.3229395683i\)
\(L(1)\) \(\approx\) \(0.4622962614 + 0.3229395683i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.0275 + 0.999i)T \)
3 \( 1 + (-0.982 + 0.186i)T \)
5 \( 1 + (-0.857 + 0.514i)T \)
7 \( 1 + (-0.768 - 0.639i)T \)
11 \( 1 + (0.999 + 0.0440i)T \)
13 \( 1 + (-0.999 - 0.0110i)T \)
17 \( 1 + (0.411 + 0.911i)T \)
19 \( 1 + (0.685 - 0.728i)T \)
23 \( 1 + (-0.518 - 0.854i)T \)
29 \( 1 + (0.975 + 0.218i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (0.815 + 0.578i)T \)
41 \( 1 + (-0.754 - 0.656i)T \)
43 \( 1 + (-0.968 + 0.250i)T \)
47 \( 1 + (-0.592 - 0.805i)T \)
53 \( 1 + (0.959 - 0.282i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (-0.381 + 0.924i)T \)
67 \( 1 + (-0.234 + 0.972i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.0935 - 0.995i)T \)
79 \( 1 + (-0.889 - 0.456i)T \)
83 \( 1 + (0.952 + 0.303i)T \)
89 \( 1 + (-0.739 + 0.673i)T \)
97 \( 1 + (-0.256 + 0.966i)T \)
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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

−22.81321399089676005575749856113, −22.11010247337061509645710891782, −21.4984623625707652993209345443, −20.21117139939279520185812988669, −19.60792563113575340669049731095, −18.85195943862927339422907182678, −18.14927979541075454756015160047, −17.02664481000000459759601839960, −16.43678801821455658750582001887, −15.45758130669853483149236469242, −14.28736192732694230725583821982, −13.13317082668087362972084915256, −12.28802106278329575800149357185, −11.862583521855895023124154696341, −11.34854910191080164492579232929, −9.77341738565957329436869739209, −9.587286941476753918241271007241, −8.1858523073131525636534896667, −7.16845732739629366263277130887, −5.866857360524628074091712752836, −5.0250461541907244019757695438, −4.089669202367626431184397829139, −3.08906262614677838004507288619, −1.6385489210601027469283478377, −0.43175366252691184772322419257, 0.881669832634960474701079196, 3.327390502311600098863970052448, 4.152259540916435143234201712284, 4.95672424263597387874998016759, 6.27770999681840619148865194145, 6.82747907631780143015052428619, 7.46879299078392543439745426746, 8.72912260379435172835737519863, 9.95611014543099553211287473366, 10.43119071855778381392889538180, 11.85345792157221358181713379829, 12.36323457758146577826760537736, 13.4890531234675303107888177130, 14.62355049345310593466919750334, 15.18998682272393727913782533616, 16.325173277591227998938471194931, 16.57681840053822252032484098110, 17.52489253251251677654054622194, 18.3102564109406416507540492178, 19.37372196871514649993001161850, 19.88397105359515961668257673989, 21.70427675010581361415060787360, 22.23984664879121713974866065357, 22.78595169845758273171743458685, 23.61647121461526314535192026665

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