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Properties

Label 1-571-571.167-r0-0-0
Degree $1$
Conductor $571$
Sign $0.987 - 0.159i$
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  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (−0.809 − 0.587i)7-s + 8-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + 16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (−0.809 − 0.587i)7-s + 8-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)10-s + (0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + 16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135485073 - 0.1713861144i\)
\(L(\frac12)\) \(\approx\) \(2.135485073 - 0.1713861144i\)
\(L(1)\) \(\approx\) \(1.604445698 - 0.1097223220i\)
\(L(1)\) \(\approx\) \(1.604445698 - 0.1097223220i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−23.19099803305749527415652942350, −22.38145571293608356154263365214, −21.739235217598035036826324893048, −21.07233623403992578971329886061, −20.12522541671295825536705833246, −19.55009440275796885634556483121, −17.877358848272150529372930159073, −17.268747942519185090253336753307, −16.306613794602767534650126109690, −15.59627289946224926160840471166, −15.17337450362685269231771771029, −13.789653631481813900639343463329, −12.74046309151810602392778935984, −12.4111158357412863670173979624, −11.59951031256648931700164743076, −10.327365545271209339905874458403, −9.76837182492349680376290513879, −8.59790558040805307953657491603, −7.17534198566519270777146722571, −6.007195635415656895299786115484, −5.64899497237175026373990950742, −4.60936440992110967767806465784, −3.847131029902624843722011250929, −2.58276437068941147775458755462, −1.13432889138474058071322079657, 1.16695051217269066207244733439, 2.504524673105575625029421789923, 3.45274790032566208157864060144, 4.51440350219862977170366426083, 5.87212433216194555177748246543, 6.37747509225970761052595444476, 6.97051724900167149752545113900, 7.99438626158053291980050159751, 9.8742217616910507381131702888, 10.54419719675122442575305977519, 11.52861398817903646912804917024, 11.997426046946403855716381131234, 13.20040103980607694581674145232, 13.94551121406681967943455742167, 14.24671149516858103261952082393, 15.88902915727210302709219599714, 16.34783729020262429338142468733, 17.13742382385939979855865591817, 18.44137765714646630459147806766, 18.990685355297444999892155861256, 19.83035271093961164019561838778, 21.09280087035322385961895650561, 21.83388513337810573992874516336, 22.62927763285207554021821189002, 22.99975133532560707339721713483

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