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Properties

Label 1-571-571.182-r0-0-0
Degree $1$
Conductor $571$
Sign $0.582 + 0.812i$
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  + (−0.986 − 0.164i)2-s + (0.991 − 0.131i)3-s + (0.945 + 0.324i)4-s + (0.746 + 0.665i)5-s + (−0.999 − 0.0330i)6-s + (0.652 + 0.757i)7-s + (−0.879 − 0.475i)8-s + (0.965 − 0.261i)9-s + (−0.627 − 0.778i)10-s + (−0.627 + 0.778i)11-s + (0.980 + 0.197i)12-s + (−0.846 − 0.533i)13-s + (−0.518 − 0.854i)14-s + (0.828 + 0.560i)15-s + (0.789 + 0.614i)16-s + (−0.340 + 0.940i)17-s + ⋯
L(s)  = 1  + (−0.986 − 0.164i)2-s + (0.991 − 0.131i)3-s + (0.945 + 0.324i)4-s + (0.746 + 0.665i)5-s + (−0.999 − 0.0330i)6-s + (0.652 + 0.757i)7-s + (−0.879 − 0.475i)8-s + (0.965 − 0.261i)9-s + (−0.627 − 0.778i)10-s + (−0.627 + 0.778i)11-s + (0.980 + 0.197i)12-s + (−0.846 − 0.533i)13-s + (−0.518 − 0.854i)14-s + (0.828 + 0.560i)15-s + (0.789 + 0.614i)16-s + (−0.340 + 0.940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ 0.582 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298001840 + 0.6667898148i\)
\(L(\frac12)\) \(\approx\) \(1.298001840 + 0.6667898148i\)
\(L(1)\) \(\approx\) \(1.098829188 + 0.2167234376i\)
\(L(1)\) \(\approx\) \(1.098829188 + 0.2167234376i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.986 - 0.164i)T \)
3 \( 1 + (0.991 - 0.131i)T \)
5 \( 1 + (0.746 + 0.665i)T \)
7 \( 1 + (0.652 + 0.757i)T \)
11 \( 1 + (-0.627 + 0.778i)T \)
13 \( 1 + (-0.846 - 0.533i)T \)
17 \( 1 + (-0.340 + 0.940i)T \)
19 \( 1 + (-0.724 + 0.689i)T \)
23 \( 1 + (0.431 - 0.901i)T \)
29 \( 1 + (0.245 - 0.969i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (0.997 + 0.0660i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (-0.934 + 0.355i)T \)
47 \( 1 + (0.789 + 0.614i)T \)
53 \( 1 + (-0.461 - 0.887i)T \)
59 \( 1 + (-0.0825 - 0.996i)T \)
61 \( 1 + (0.894 - 0.446i)T \)
67 \( 1 + (-0.461 - 0.887i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.997 + 0.0660i)T \)
79 \( 1 + (-0.574 + 0.818i)T \)
83 \( 1 + (-0.999 - 0.0330i)T \)
89 \( 1 + (-0.999 - 0.0330i)T \)
97 \( 1 + (0.601 - 0.799i)T \)
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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

−23.819812868630347170730837429235, −21.74073743212449707499348764436, −21.23979271601967092944043010602, −20.4415996175269577141525158421, −19.87909873844067598824995939716, −18.98821666736137225885103794599, −18.1072177538028694426713351255, −17.22828338001396955163769900062, −16.563404527371539951949064372042, −15.67088659366895728146859930941, −14.72474248989528732271300082288, −13.81593912597517714418587484502, −13.228011563214351952279086031246, −11.830787812874249918633061372214, −10.74585291786988213225235127791, −9.98702217283729597219150196163, −9.10645921519711692164633682954, −8.54728099976541313502368506629, −7.55844785252388543641509787393, −6.849594169758590691442207453568, −5.37183668838086961810449113682, −4.46203419440276981970921425404, −2.84926875760177239520908711071, −2.03840150341870801029384000075, −0.91943582142431343810896778026, 1.66657248967396256802039163881, 2.336269370875779117970198252059, 2.971474055413974318042363296463, 4.59930155601857771925267631124, 6.06600040104868972927614380755, 6.95000193729982361784302486436, 8.04134465023566535167400323198, 8.437744997757268665776176726801, 9.73959627849041224533487782274, 10.08529359449132407992440464563, 11.08238799431709610369645202132, 12.42283190570933658614471890511, 12.946432911026534958194689621597, 14.42417152108943060972664731133, 14.998502174071323112380245768327, 15.509075616855386567647374187704, 17.02884329035143052003428267225, 17.74461489751188851639537941012, 18.42820191111024245546076590059, 19.05705163401781381058125307690, 19.93730631805912659948279288253, 20.91456038311866536966449223055, 21.30680077543638452705399446268, 22.19519534102013612095859939804, 23.62812827429345695425760905299

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