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Properties

Label 1-571-571.132-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.632 + 0.774i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (−0.789 + 0.614i)2-s + (−0.746 + 0.665i)3-s + (0.245 − 0.969i)4-s + (0.652 + 0.757i)5-s + (0.180 − 0.983i)6-s + (0.0165 − 0.999i)7-s + (0.401 + 0.915i)8-s + (0.115 − 0.993i)9-s + (−0.980 − 0.197i)10-s + (0.980 − 0.197i)11-s + (0.461 + 0.887i)12-s + (0.0495 + 0.998i)13-s + (0.601 + 0.799i)14-s + (−0.991 − 0.131i)15-s + (−0.879 − 0.475i)16-s + (−0.431 − 0.901i)17-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)2-s + (−0.746 + 0.665i)3-s + (0.245 − 0.969i)4-s + (0.652 + 0.757i)5-s + (0.180 − 0.983i)6-s + (0.0165 − 0.999i)7-s + (0.401 + 0.915i)8-s + (0.115 − 0.993i)9-s + (−0.980 − 0.197i)10-s + (0.980 − 0.197i)11-s + (0.461 + 0.887i)12-s + (0.0495 + 0.998i)13-s + (0.601 + 0.799i)14-s + (−0.991 − 0.131i)15-s + (−0.879 − 0.475i)16-s + (−0.431 − 0.901i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3658493452 + 0.7709374861i\)
\(L(\frac12)\) \(\approx\) \(0.3658493452 + 0.7709374861i\)
\(L(1)\) \(\approx\) \(0.5711295068 + 0.2961623722i\)
\(L(1)\) \(\approx\) \(0.5711295068 + 0.2961623722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.789 + 0.614i)T \)
3 \( 1 + (-0.746 + 0.665i)T \)
5 \( 1 + (0.652 + 0.757i)T \)
7 \( 1 + (0.0165 - 0.999i)T \)
11 \( 1 + (0.980 - 0.197i)T \)
13 \( 1 + (0.0495 + 0.998i)T \)
17 \( 1 + (-0.431 - 0.901i)T \)
19 \( 1 + (-0.863 - 0.504i)T \)
23 \( 1 + (-0.995 - 0.0990i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (-0.934 - 0.355i)T \)
41 \( 1 + (0.0825 + 0.996i)T \)
43 \( 1 + (-0.909 + 0.416i)T \)
47 \( 1 + (0.879 + 0.475i)T \)
53 \( 1 + (0.277 + 0.960i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (0.828 + 0.560i)T \)
67 \( 1 + (0.277 + 0.960i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.934 + 0.355i)T \)
79 \( 1 + (0.846 - 0.533i)T \)
83 \( 1 + (0.180 - 0.983i)T \)
89 \( 1 + (-0.180 + 0.983i)T \)
97 \( 1 + (0.371 + 0.928i)T \)
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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

−22.24964575250736188525596811273, −22.04444167205710522385924289542, −21.09008270297113477114827868511, −20.01218755497676489251071135432, −19.41356139668361231464830298842, −18.39552726238979311275301341525, −17.743383182514932236004066385868, −17.190678947454493323642854865323, −16.4214290591635261975423121855, −15.431706908978588249284977725, −14.00414682788824338987282723395, −12.67109171210242754188604216844, −12.60758896800756359118622602162, −11.738511021372632762352444913420, −10.63689124954945162144213382247, −9.9031526347367247974283967660, −8.616828207558469668832880997524, −8.37943945270684356805613459094, −6.88738432352755957218937918426, −6.04386957946547683301726638950, −5.12614639434304209930739132159, −3.740555824711043866291989911922, −2.08102937213574862550848006081, −1.68744977826769533879543522449, −0.386922235268776237229956519674, 0.855940387569515838902615983073, 2.1452434637378492281257115897, 3.871495754493625752560025397835, 4.75714874688458559238984763108, 6.08831838417617053033759162394, 6.59160222927356815148787061763, 7.30412719353175200670318375711, 8.86795172851562919476667609691, 9.59923379479803752950158213159, 10.2960316664127028374312461660, 11.1213240701550940761611043777, 11.70211189875808406593408867812, 13.54527264294878278557611994281, 14.25335536333908980546789810907, 14.97105128666164629867622984264, 16.08606067199666074564341424630, 16.72689895230141276900118536150, 17.40287341473691209770010263541, 17.99278179124264052311024609325, 18.98963109302919426383356117899, 19.89745480642726215972455592386, 20.851614868491312468327336116, 21.83407643487513097965428403049, 22.58250376545738387173692544988, 23.38902180211932216365482069243

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