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Properties

Label 1-571-571.16-r0-0-0
Degree $1$
Conductor $571$
Sign $0.533 - 0.845i$
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  + (−0.754 + 0.656i)2-s + (0.451 − 0.892i)3-s + (0.137 − 0.990i)4-s + (0.975 − 0.218i)5-s + (0.245 + 0.969i)6-s + (−0.986 + 0.164i)7-s + (0.546 + 0.837i)8-s + (−0.592 − 0.805i)9-s + (−0.592 + 0.805i)10-s + (0.993 − 0.110i)11-s + (−0.821 − 0.569i)12-s + (0.0275 + 0.999i)13-s + (0.635 − 0.771i)14-s + (0.245 − 0.969i)15-s + (−0.962 − 0.272i)16-s + (−0.962 − 0.272i)17-s + ⋯
L(s)  = 1  + (−0.754 + 0.656i)2-s + (0.451 − 0.892i)3-s + (0.137 − 0.990i)4-s + (0.975 − 0.218i)5-s + (0.245 + 0.969i)6-s + (−0.986 + 0.164i)7-s + (0.546 + 0.837i)8-s + (−0.592 − 0.805i)9-s + (−0.592 + 0.805i)10-s + (0.993 − 0.110i)11-s + (−0.821 − 0.569i)12-s + (0.0275 + 0.999i)13-s + (0.635 − 0.771i)14-s + (0.245 − 0.969i)15-s + (−0.962 − 0.272i)16-s + (−0.962 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.533 - 0.845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027079311 - 0.5662128148i\)
\(L(\frac12)\) \(\approx\) \(1.027079311 - 0.5662128148i\)
\(L(1)\) \(\approx\) \(0.9351582115 - 0.1677341322i\)
\(L(1)\) \(\approx\) \(0.9351582115 - 0.1677341322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.754 + 0.656i)T \)
3 \( 1 + (0.451 - 0.892i)T \)
5 \( 1 + (0.975 - 0.218i)T \)
7 \( 1 + (-0.986 + 0.164i)T \)
11 \( 1 + (0.993 - 0.110i)T \)
13 \( 1 + (0.0275 + 0.999i)T \)
17 \( 1 + (-0.962 - 0.272i)T \)
19 \( 1 + (0.451 - 0.892i)T \)
23 \( 1 + (0.546 - 0.837i)T \)
29 \( 1 + (0.851 - 0.523i)T \)
31 \( 1 + (0.245 + 0.969i)T \)
37 \( 1 + (0.0275 - 0.999i)T \)
41 \( 1 + (0.975 - 0.218i)T \)
43 \( 1 + (-0.592 + 0.805i)T \)
47 \( 1 + (0.716 - 0.697i)T \)
53 \( 1 + (-0.754 - 0.656i)T \)
59 \( 1 + (-0.986 + 0.164i)T \)
61 \( 1 + (-0.191 - 0.981i)T \)
67 \( 1 + (-0.191 + 0.981i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.851 + 0.523i)T \)
79 \( 1 + (-0.926 - 0.376i)T \)
83 \( 1 + (0.716 - 0.697i)T \)
89 \( 1 + (-0.962 - 0.272i)T \)
97 \( 1 + (0.137 - 0.990i)T \)
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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

−22.7750498068496374628486119300, −22.27239966935888199066666618881, −21.71446697597682708105971971793, −20.718712851479178064388662853889, −20.07471798284662012552682715617, −19.43720623899728457624756385564, −18.46984174845351872539782497661, −17.341376998159759525270479734959, −16.95714492474716888753731526756, −15.935408743395689700090629690269, −15.10482072045536306014563413065, −13.86879878800492055406876729089, −13.24467870588255542161778039183, −12.21649272440352932111268397378, −10.95942583415564685962601553235, −10.315667587217646313891063111001, −9.53230176989003626169906120957, −9.13342006276213009237178752977, −8.00368218703885295388919551470, −6.78185276024141248011023978448, −5.78155081739649752899157934255, −4.32190510951736492274276110135, −3.31675162280534413492610470116, −2.668143710533542975100085043111, −1.35872420340872321901539987188, 0.79492330329961873936561555919, 1.918214324789575940865641974045, 2.82322176601062142405545782091, 4.59104443335008225207116512039, 5.981843541407122715149114933911, 6.64457349325256982738252858790, 7.0228851498736593407222474927, 8.623791376107945157603357763523, 9.1073230603316727805581952659, 9.650097725169622435213100524713, 10.98783310594001152118919911215, 12.09477050836707783699663901747, 13.16899444204010563847801660950, 13.91242691774578053146966584792, 14.478888743607306521734532078349, 15.70157171268609398794284444008, 16.562547616583832393431266232817, 17.39214978233771763420470768408, 18.00885688974049704025590419597, 18.91432066816518852567059667682, 19.59199120359340578584232090263, 20.17481145972521808556869833266, 21.43683576631272502780537697980, 22.47520550007227969872103659392, 23.32693940031199881887613331803

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