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Properties

Label 1-571-571.361-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.206 + 0.978i$
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 + (−0.795 − 0.605i)3-s + (0.451 − 0.892i)4-s + (−0.421 − 0.906i)5-s + (−0.995 − 0.0990i)6-s + (−0.846 + 0.533i)7-s + (−0.0825 − 0.996i)8-s + (0.266 + 0.963i)9-s + (−0.834 − 0.551i)10-s + (−0.0605 − 0.998i)11-s + (−0.899 + 0.436i)12-s + (−0.917 + 0.396i)13-s + (−0.441 + 0.897i)14-s + (−0.213 + 0.976i)15-s + (−0.592 − 0.805i)16-s + (0.00551 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.851 − 0.523i)2-s + (−0.795 − 0.605i)3-s + (0.451 − 0.892i)4-s + (−0.421 − 0.906i)5-s + (−0.995 − 0.0990i)6-s + (−0.846 + 0.533i)7-s + (−0.0825 − 0.996i)8-s + (0.266 + 0.963i)9-s + (−0.834 − 0.551i)10-s + (−0.0605 − 0.998i)11-s + (−0.899 + 0.436i)12-s + (−0.917 + 0.396i)13-s + (−0.441 + 0.897i)14-s + (−0.213 + 0.976i)15-s + (−0.592 − 0.805i)16-s + (0.00551 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.206 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2327572583 - 0.2870110030i\)
\(L(\frac12)\) \(\approx\) \(-0.2327572583 - 0.2870110030i\)
\(L(1)\) \(\approx\) \(0.6200409770 - 0.5734730102i\)
\(L(1)\) \(\approx\) \(0.6200409770 - 0.5734730102i\)

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 \( 1 + (-0.795 - 0.605i)T \)
5 \( 1 + (-0.421 - 0.906i)T \)
7 \( 1 + (-0.846 + 0.533i)T \)
11 \( 1 + (-0.0605 - 0.998i)T \)
13 \( 1 + (-0.917 + 0.396i)T \)
17 \( 1 + (0.00551 + 0.999i)T \)
19 \( 1 + (0.329 - 0.944i)T \)
23 \( 1 + (-0.973 + 0.229i)T \)
29 \( 1 + (-0.298 - 0.954i)T \)
31 \( 1 + (-0.401 + 0.915i)T \)
37 \( 1 + (-0.660 + 0.750i)T \)
41 \( 1 + (-0.191 + 0.981i)T \)
43 \( 1 + (0.999 - 0.0440i)T \)
47 \( 1 + (0.993 - 0.110i)T \)
53 \( 1 + (-0.381 - 0.924i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (-0.942 + 0.335i)T \)
67 \( 1 + (-0.609 + 0.792i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.319 - 0.947i)T \)
79 \( 1 + (-0.256 - 0.966i)T \)
83 \( 1 + (0.411 + 0.911i)T \)
89 \( 1 + (0.583 - 0.812i)T \)
97 \( 1 + (0.159 + 0.987i)T \)
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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

−23.48867977072195722480632587560, −22.78363411338352442685974292783, −22.48004128396864758125552778521, −21.87113213956460739943780066456, −20.51445805083966270111171747137, −20.11520501370504394375975745461, −18.660941900883349607271765688409, −17.7273385091563500900986780224, −16.951218137297122393923339352022, −16.03671671859992312518650087365, −15.55438799348207211006120240295, −14.62133520498415757696058762302, −13.95385136357926019616255586209, −12.39295471665386567999290231541, −12.29541672601391771405139884394, −11.03792530410747952127944715940, −10.24681482268539708367527344777, −9.41861663659447272239430558552, −7.514824427327142095337239589963, −7.20050791457092948882438156495, −6.19480484558818722870878769133, −5.296745489018869296625539543523, −4.205740832917180066829027586, −3.56427377726245663581396733192, −2.45816007111081532251568976288, 0.15245142796878243341500708649, 1.47135353762737738207334674995, 2.64267522773702774856446030996, 3.88391990045559635153739757822, 4.9278309245033049340642613175, 5.732518590483886381532164467961, 6.43897572973727171080669050994, 7.57252531856756816644680873124, 8.8374809140951523438810533211, 9.88938533284993221725182991259, 10.943438532517599799675974402175, 11.94739116863458888962716805030, 12.21443632364896097667334004327, 13.186627074223084767220556830651, 13.686834173490994018791990822011, 15.10151794013113667455760125937, 16.015515775337141701902174113745, 16.53079683485888572875184022630, 17.60744924937563022593178107524, 18.91220764524018582378514237099, 19.363364566729910272149602383625, 19.95942079766360588268731444894, 21.34307370313060173884181387899, 21.89998357320428936157397765285, 22.54915984859734971707521272773

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