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Properties

Label 1-571-571.204-r0-0-0
Degree $1$
Conductor $571$
Sign $0.815 - 0.579i$
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  + (0.975 + 0.218i)2-s + (−0.644 − 0.764i)3-s + (0.904 + 0.426i)4-s + (−0.997 + 0.0770i)5-s + (−0.461 − 0.887i)6-s + (−0.995 − 0.0990i)7-s + (0.789 + 0.614i)8-s + (−0.170 + 0.985i)9-s + (−0.989 − 0.142i)10-s + (0.618 + 0.785i)11-s + (−0.256 − 0.966i)12-s + (0.224 − 0.974i)13-s + (−0.949 − 0.314i)14-s + (0.701 + 0.712i)15-s + (0.635 + 0.771i)16-s + (−0.0605 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.975 + 0.218i)2-s + (−0.644 − 0.764i)3-s + (0.904 + 0.426i)4-s + (−0.997 + 0.0770i)5-s + (−0.461 − 0.887i)6-s + (−0.995 − 0.0990i)7-s + (0.789 + 0.614i)8-s + (−0.170 + 0.985i)9-s + (−0.989 − 0.142i)10-s + (0.618 + 0.785i)11-s + (−0.256 − 0.966i)12-s + (0.224 − 0.974i)13-s + (−0.949 − 0.314i)14-s + (0.701 + 0.712i)15-s + (0.635 + 0.771i)16-s + (−0.0605 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.815 - 0.579i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.815 - 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530335732 - 0.4882084853i\)
\(L(\frac12)\) \(\approx\) \(1.530335732 - 0.4882084853i\)
\(L(1)\) \(\approx\) \(1.292350951 - 0.1536472134i\)
\(L(1)\) \(\approx\) \(1.292350951 - 0.1536472134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.975 + 0.218i)T \)
3 \( 1 + (-0.644 - 0.764i)T \)
5 \( 1 + (-0.997 + 0.0770i)T \)
7 \( 1 + (-0.995 - 0.0990i)T \)
11 \( 1 + (0.618 + 0.785i)T \)
13 \( 1 + (0.224 - 0.974i)T \)
17 \( 1 + (-0.0605 - 0.998i)T \)
19 \( 1 + (0.528 - 0.849i)T \)
23 \( 1 + (0.828 + 0.560i)T \)
29 \( 1 + (-0.191 - 0.981i)T \)
31 \( 1 + (-0.986 + 0.164i)T \)
37 \( 1 + (0.996 + 0.0880i)T \)
41 \( 1 + (0.851 + 0.523i)T \)
43 \( 1 + (0.884 - 0.466i)T \)
47 \( 1 + (0.350 - 0.936i)T \)
53 \( 1 + (-0.917 - 0.396i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (0.815 - 0.578i)T \)
67 \( 1 + (0.802 - 0.596i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.421 - 0.906i)T \)
79 \( 1 + (0.287 - 0.957i)T \)
83 \( 1 + (0.999 + 0.0440i)T \)
89 \( 1 + (-0.537 + 0.843i)T \)
97 \( 1 + (-0.982 + 0.186i)T \)
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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

−23.30865926370287849000524534632, −22.30291856590471675094297954042, −22.02591180777588262103019454436, −21.00091517748245495482265359419, −20.18319111212167144288388934550, −19.27849922702062487905359301951, −18.72613170262391539312003162345, −16.91848489765546603433812368009, −16.27742750742234040526452275073, −15.94719600996157625880149258986, −14.82417377833197384207606361726, −14.25186595867384123400998067827, −12.745943056976674659499972191348, −12.36486250735455994613983750344, −11.20731893127414363001144380462, −10.96440269270149545978651591914, −9.67273871749540856808291156018, −8.767540541200584590518270496956, −7.21256101484818288807140648681, −6.319165264438080580244515144699, −5.649292978169556397952321465340, −4.31442000702514848266038673021, −3.81423301888563218872006168973, −3.03624454146811884971579373352, −1.1121097979344244466935126975, 0.80995741269109877869633657378, 2.51894317864675513704239057659, 3.41640995537387738858784865453, 4.53694035031808894692710758209, 5.473275567552417801974322822583, 6.518871282634463980184605780684, 7.25379027058656462689453214378, 7.73975753038579533001554157429, 9.30537175553995960083577955713, 10.729694356289798217677869649212, 11.49169244871720361000882712224, 12.186514976658920242337083139724, 12.94900962641706293858385905841, 13.52812771193780530247079715971, 14.78104856795628244522060912523, 15.63090668658739440228248581773, 16.24044647563694323121353517606, 17.14624275459839075902048582122, 18.08094270067074023047614333238, 19.24953267376427884274569845608, 19.8612580925589547809070811710, 20.50787583692916472038619623277, 22.12108687328491014385866744648, 22.508143235957019419154946076404, 23.16937213865670475500579748094

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