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Properties

Label 1-571-571.309-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.0759 + 0.997i$
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  + (0.716 + 0.697i)2-s + (0.975 + 0.218i)3-s + (0.0275 + 0.999i)4-s + (0.350 − 0.936i)5-s + (0.546 + 0.837i)6-s + (0.789 + 0.614i)7-s + (−0.677 + 0.735i)8-s + (0.904 + 0.426i)9-s + (0.904 − 0.426i)10-s + (−0.821 + 0.569i)11-s + (−0.191 + 0.981i)12-s + (−0.592 + 0.805i)13-s + (0.137 + 0.990i)14-s + (0.546 − 0.837i)15-s + (−0.998 + 0.0550i)16-s + (−0.998 + 0.0550i)17-s + ⋯
L(s)  = 1  + (0.716 + 0.697i)2-s + (0.975 + 0.218i)3-s + (0.0275 + 0.999i)4-s + (0.350 − 0.936i)5-s + (0.546 + 0.837i)6-s + (0.789 + 0.614i)7-s + (−0.677 + 0.735i)8-s + (0.904 + 0.426i)9-s + (0.904 − 0.426i)10-s + (−0.821 + 0.569i)11-s + (−0.191 + 0.981i)12-s + (−0.592 + 0.805i)13-s + (0.137 + 0.990i)14-s + (0.546 − 0.837i)15-s + (−0.998 + 0.0550i)16-s + (−0.998 + 0.0550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0759 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0759 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.0759 + 0.997i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.0759 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.043676420 + 2.205288607i\)
\(L(\frac12)\) \(\approx\) \(2.043676420 + 2.205288607i\)
\(L(1)\) \(\approx\) \(1.839103209 + 1.101764481i\)
\(L(1)\) \(\approx\) \(1.839103209 + 1.101764481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.716 + 0.697i)T \)
3 \( 1 + (0.975 + 0.218i)T \)
5 \( 1 + (0.350 - 0.936i)T \)
7 \( 1 + (0.789 + 0.614i)T \)
11 \( 1 + (-0.821 + 0.569i)T \)
13 \( 1 + (-0.592 + 0.805i)T \)
17 \( 1 + (-0.998 + 0.0550i)T \)
19 \( 1 + (0.975 + 0.218i)T \)
23 \( 1 + (-0.677 - 0.735i)T \)
29 \( 1 + (0.993 + 0.110i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (-0.592 - 0.805i)T \)
41 \( 1 + (0.350 - 0.936i)T \)
43 \( 1 + (0.904 - 0.426i)T \)
47 \( 1 + (0.451 - 0.892i)T \)
53 \( 1 + (0.716 - 0.697i)T \)
59 \( 1 + (0.789 + 0.614i)T \)
61 \( 1 + (-0.962 + 0.272i)T \)
67 \( 1 + (-0.962 - 0.272i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.993 - 0.110i)T \)
79 \( 1 + (0.851 + 0.523i)T \)
83 \( 1 + (0.451 - 0.892i)T \)
89 \( 1 + (-0.998 + 0.0550i)T \)
97 \( 1 + (0.0275 + 0.999i)T \)
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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

−22.90318734201659366722803755841, −22.02882593588009478703953522271, −21.32724952841047722437465090571, −20.55286095331848758605315675854, −19.86106002688951444549627920452, −19.10610477123657138912478673594, −18.10896822591872393487928094672, −17.69891750932363065274129957185, −15.73447463299624853285503011845, −15.20718733493608900548806853934, −14.2609728182099938935721730422, −13.7031082806580776619796353230, −13.20184820492096729322849345531, −11.90892240264975138081495715653, −10.95972137418237419787311926257, −10.24503443901842649296811533173, −9.489218890199566570386007118420, −8.07343379293455607479386419145, −7.33805866956218465931131219811, −6.22480684253236352546958418505, −5.06584810736312698433694893336, −4.02056970943045146306176000225, −2.91077519349569274364897323114, −2.45286317452494121237286804605, −1.16513392488946995437246655038, 1.96258232102356935700501263021, 2.56191173849991958628469634621, 4.14435113575312826105653334285, 4.768304101452008591275717173985, 5.49049930354979152700717489879, 6.92483235090072067135493007255, 7.84005833062244480640244773610, 8.64411622581120834124131643532, 9.1980756533646497474412774098, 10.44848186120659017053306237606, 12.06391463649505775864134985315, 12.437966555672486554060832151719, 13.65805642514038876324892944756, 14.03934865468876483745765369922, 15.03204754837861928724105449260, 15.78598715990029614609596689158, 16.364060563247966335937965618942, 17.65482680704350178289051927630, 18.13205877899708747189950092572, 19.57797898982288047939005395027, 20.49373424023164251822617483022, 21.06879095884245533409960755787, 21.62012695634202818865304154723, 22.53825819947801358020696338091, 23.94579506016736481573898822416

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