Properties

Label 1-571-571.126-r0-0-0
Degree $1$
Conductor $571$
Sign $0.105 - 0.994i$
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 + (−0.660 + 0.750i)3-s + (0.0275 − 0.999i)4-s + (0.999 − 0.0440i)5-s + (0.0495 + 0.998i)6-s + (−0.277 + 0.960i)7-s + (−0.677 − 0.735i)8-s + (−0.126 − 0.991i)9-s + (0.685 − 0.728i)10-s + (0.287 − 0.957i)11-s + (0.731 + 0.681i)12-s + (−0.949 + 0.314i)13-s + (0.471 + 0.882i)14-s + (−0.627 + 0.778i)15-s + (−0.998 − 0.0550i)16-s + (−0.256 − 0.966i)17-s + ⋯
L(s)  = 1  + (0.716 − 0.697i)2-s + (−0.660 + 0.750i)3-s + (0.0275 − 0.999i)4-s + (0.999 − 0.0440i)5-s + (0.0495 + 0.998i)6-s + (−0.277 + 0.960i)7-s + (−0.677 − 0.735i)8-s + (−0.126 − 0.991i)9-s + (0.685 − 0.728i)10-s + (0.287 − 0.957i)11-s + (0.731 + 0.681i)12-s + (−0.949 + 0.314i)13-s + (0.471 + 0.882i)14-s + (−0.627 + 0.778i)15-s + (−0.998 − 0.0550i)16-s + (−0.256 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.105 - 0.994i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237313074 - 1.113442676i\)
\(L(\frac12)\) \(\approx\) \(1.237313074 - 1.113442676i\)
\(L(1)\) \(\approx\) \(1.245127498 - 0.4813952077i\)
\(L(1)\) \(\approx\) \(1.245127498 - 0.4813952077i\)

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 \( 1 + (-0.660 + 0.750i)T \)
5 \( 1 + (0.999 - 0.0440i)T \)
7 \( 1 + (-0.277 + 0.960i)T \)
11 \( 1 + (0.287 - 0.957i)T \)
13 \( 1 + (-0.949 + 0.314i)T \)
17 \( 1 + (-0.256 - 0.966i)T \)
19 \( 1 + (0.0935 - 0.995i)T \)
23 \( 1 + (0.115 - 0.993i)T \)
29 \( 1 + (0.993 - 0.110i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (0.583 + 0.812i)T \)
41 \( 1 + (0.350 + 0.936i)T \)
43 \( 1 + (-0.480 - 0.876i)T \)
47 \( 1 + (0.451 + 0.892i)T \)
53 \( 1 + (-0.441 + 0.897i)T \)
59 \( 1 + (0.789 - 0.614i)T \)
61 \( 1 + (0.938 - 0.345i)T \)
67 \( 1 + (-0.556 - 0.831i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.411 - 0.911i)T \)
79 \( 1 + (-0.381 + 0.924i)T \)
83 \( 1 + (-0.889 - 0.456i)T \)
89 \( 1 + (0.840 - 0.542i)T \)
97 \( 1 + (-0.942 - 0.335i)T \)
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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

−23.329721970609364213200138458052, −22.835039364863608526778210224352, −22.02763647437660063357626185503, −21.28600192072088574697829022853, −20.15979769925499470025211395527, −19.324711126118016747907788265497, −17.76305495109695926889920825827, −17.586609240793129805732401837923, −16.90260994401267140319776004728, −16.05174228929540234461372358888, −14.69887322878666440934768143719, −14.11890698180520118570258402104, −13.1635657464485127973073284101, −12.6841355111136847012867700730, −11.84215913150361748888128501128, −10.52626354431411552045273225311, −9.80277747881943216653364621155, −8.30935442183105893940002482244, −7.261622330188524759017975801769, −6.78740438007545843378896846620, −5.85019009151416499356862333036, −5.0513044554355679418635719609, −4.01521124444487996976036550866, −2.56168190572992410040815153528, −1.46228682058107087467209127226, 0.75754546836245928229718880295, 2.45748483109456162534347541177, 2.99992064519617630845628856739, 4.60155287235739243471882615936, 5.08011901314456564296755339924, 6.13344377280222101591296323672, 6.57111295408471644310938560241, 8.87081728424730111745704242049, 9.45522563061925661694509711603, 10.16117251695103396241654373097, 11.22097072912738875919014151415, 11.83418625376849943921466312003, 12.70804416534742068046984701577, 13.70495053080317142736298947966, 14.50523834868642493113637773593, 15.37950694000681498151632589982, 16.22162516340976322401817997393, 17.15726297617568250064846976743, 18.165330490272391027573780752035, 18.872909017355869256102718084, 19.9734016986034436979929508182, 20.93615445759668946610312281419, 21.56303307467137721927620923066, 22.24363561643832774850392060799, 22.41155064695133020786880814649

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