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Properties

Label 1-571-571.4-r0-0-0
Degree $1$
Conductor $571$
Sign $0.993 + 0.113i$
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  + (0.350 + 0.936i)2-s + (0.851 − 0.523i)3-s + (−0.754 + 0.656i)4-s + (0.993 − 0.110i)5-s + (0.789 + 0.614i)6-s + (−0.0825 − 0.996i)7-s + (−0.879 − 0.475i)8-s + (0.451 − 0.892i)9-s + (0.451 + 0.892i)10-s + (−0.998 + 0.0550i)11-s + (−0.298 + 0.954i)12-s + (0.716 + 0.697i)13-s + (0.904 − 0.426i)14-s + (0.789 − 0.614i)15-s + (0.137 − 0.990i)16-s + (0.137 − 0.990i)17-s + ⋯
L(s)  = 1  + (0.350 + 0.936i)2-s + (0.851 − 0.523i)3-s + (−0.754 + 0.656i)4-s + (0.993 − 0.110i)5-s + (0.789 + 0.614i)6-s + (−0.0825 − 0.996i)7-s + (−0.879 − 0.475i)8-s + (0.451 − 0.892i)9-s + (0.451 + 0.892i)10-s + (−0.998 + 0.0550i)11-s + (−0.298 + 0.954i)12-s + (0.716 + 0.697i)13-s + (0.904 − 0.426i)14-s + (0.789 − 0.614i)15-s + (0.137 − 0.990i)16-s + (0.137 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.993 + 0.113i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.993 + 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.329575391 + 0.1331053796i\)
\(L(\frac12)\) \(\approx\) \(2.329575391 + 0.1331053796i\)
\(L(1)\) \(\approx\) \(1.692447328 + 0.2681337408i\)
\(L(1)\) \(\approx\) \(1.692447328 + 0.2681337408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.350 + 0.936i)T \)
3 \( 1 + (0.851 - 0.523i)T \)
5 \( 1 + (0.993 - 0.110i)T \)
7 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (-0.998 + 0.0550i)T \)
13 \( 1 + (0.716 + 0.697i)T \)
17 \( 1 + (0.137 - 0.990i)T \)
19 \( 1 + (0.851 - 0.523i)T \)
23 \( 1 + (-0.879 + 0.475i)T \)
29 \( 1 + (-0.962 + 0.272i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.716 - 0.697i)T \)
41 \( 1 + (0.993 - 0.110i)T \)
43 \( 1 + (0.451 + 0.892i)T \)
47 \( 1 + (-0.926 + 0.376i)T \)
53 \( 1 + (0.350 - 0.936i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (0.635 - 0.771i)T \)
67 \( 1 + (0.635 + 0.771i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.962 - 0.272i)T \)
79 \( 1 + (-0.191 + 0.981i)T \)
83 \( 1 + (-0.926 + 0.376i)T \)
89 \( 1 + (0.137 - 0.990i)T \)
97 \( 1 + (-0.754 + 0.656i)T \)
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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

−22.797955560285660308715963367753, −22.118632102012240940876727016722, −21.4369473581495138525139027062, −20.797579764301209699663568542, −20.2796571587749770725433203345, −19.041831106630699048334362520620, −18.4723191083018562137561721173, −17.79729307297044295091785616855, −16.333964259886848869440498939723, −15.283899972487503306029567254675, −14.74699523346946018371076625400, −13.67216973771784662042550705650, −13.185223748433228729295426954766, −12.329532438552262420798628319600, −11.0088583887956687737637652933, −10.19762990570484073564953222681, −9.67877190547107810835361237922, −8.68220878719312850571572814817, −7.97679642130121627392079828223, −5.88297400944854192388381456137, −5.58216831963529679264948394550, −4.29896584998286676734711089136, −3.13056628927879330686744375615, −2.49864930900325846124570771110, −1.59515406100041286516729856878, 1.065886937484256263682136167417, 2.521002099729599316138043653447, 3.53496751816961969287979430301, 4.64781390705630924323806625653, 5.72807002476977925680997064979, 6.74014901852554768800972648146, 7.41892411617678405927580880615, 8.24977833300345672330731689799, 9.37899011244683412486055911547, 9.84200317221767140519039049686, 11.3824246787213371166541305606, 12.85965868895137829513595800881, 13.26754314624700556101679602413, 14.06481252286225925649098323162, 14.36702808638611398020846330146, 15.88465465779886708263793191288, 16.24405742004298510151497495679, 17.620996567672275206957135094859, 17.99027938201323518638754775507, 18.83909934740167700426523595135, 20.14374917698649263007868260960, 20.88318747676795090464673852665, 21.469427324124709299340928846568, 22.69234202572165614258877168853, 23.49810698696551638154163956606

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