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Properties

Label 1-571-571.113-r1-0-0
Degree $1$
Conductor $571$
Sign $-0.903 + 0.429i$
Analytic cond. $61.3624$
Root an. cond. $61.3624$
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  + (0.926 + 0.376i)2-s + (−0.202 − 0.979i)3-s + (0.716 + 0.697i)4-s + (0.329 − 0.944i)5-s + (0.180 − 0.983i)6-s + (0.0165 − 0.999i)7-s + (0.401 + 0.915i)8-s + (−0.917 + 0.396i)9-s + (0.660 − 0.750i)10-s + (−0.319 + 0.947i)11-s + (0.537 − 0.843i)12-s + (−0.889 − 0.456i)13-s + (0.391 − 0.920i)14-s + (−0.991 − 0.131i)15-s + (0.0275 + 0.999i)16-s + (−0.565 + 0.824i)17-s + ⋯
L(s)  = 1  + (0.926 + 0.376i)2-s + (−0.202 − 0.979i)3-s + (0.716 + 0.697i)4-s + (0.329 − 0.944i)5-s + (0.180 − 0.983i)6-s + (0.0165 − 0.999i)7-s + (0.401 + 0.915i)8-s + (−0.917 + 0.396i)9-s + (0.660 − 0.750i)10-s + (−0.319 + 0.947i)11-s + (0.537 − 0.843i)12-s + (−0.889 − 0.456i)13-s + (0.391 − 0.920i)14-s + (−0.991 − 0.131i)15-s + (0.0275 + 0.999i)16-s + (−0.565 + 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.903 + 0.429i$
Analytic conductor: \(61.3624\)
Root analytic conductor: \(61.3624\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (1:\ ),\ -0.903 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1129453845 - 0.5007900806i\)
\(L(\frac12)\) \(\approx\) \(-0.1129453845 - 0.5007900806i\)
\(L(1)\) \(\approx\) \(1.275014134 - 0.3477391881i\)
\(L(1)\) \(\approx\) \(1.275014134 - 0.3477391881i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (0.926 + 0.376i)T \)
3 \( 1 + (-0.202 - 0.979i)T \)
5 \( 1 + (0.329 - 0.944i)T \)
7 \( 1 + (0.0165 - 0.999i)T \)
11 \( 1 + (-0.319 + 0.947i)T \)
13 \( 1 + (-0.889 - 0.456i)T \)
17 \( 1 + (-0.565 + 0.824i)T \)
19 \( 1 + (0.868 - 0.495i)T \)
23 \( 1 + (-0.995 - 0.0990i)T \)
29 \( 1 + (-0.998 - 0.0550i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.159 + 0.987i)T \)
41 \( 1 + (0.821 - 0.569i)T \)
43 \( 1 + (0.0935 - 0.995i)T \)
47 \( 1 + (-0.851 + 0.523i)T \)
53 \( 1 + (-0.970 - 0.240i)T \)
59 \( 1 + (0.945 + 0.324i)T \)
61 \( 1 + (-0.899 + 0.436i)T \)
67 \( 1 + (0.693 - 0.720i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.775 + 0.631i)T \)
79 \( 1 + (0.0385 + 0.999i)T \)
83 \( 1 + (0.761 + 0.648i)T \)
89 \( 1 + (0.942 - 0.335i)T \)
97 \( 1 + (-0.989 - 0.142i)T \)
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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

−23.13407213767921819913136406083, −22.379503977930238850427659002165, −21.81766790652653144423647381498, −21.48324237360296618862757997812, −20.4611346988787773373768913802, −19.5162496563017427500806892125, −18.59029441757688937177877145662, −17.81920538994318584391094499767, −16.21120145924085865435873171148, −15.99889050821300388561062931039, −14.73422904049611450769882390319, −14.48868645090850509540498676663, −13.518858910335927715513178802789, −12.23249876550283859518316615629, −11.401651296322270510534129014912, −10.94922141842914459560176725421, −9.77573777129716613665304144785, −9.2995787559805498779378419504, −7.65846944930351919402098317968, −6.370166835555699205665133876959, −5.662685445246873514789425827915, −4.965414935105497607684299538721, −3.6733218228356235866198044826, −2.91031862749661179995346958939, −2.08398105913014872931733577993, 0.07975526942223633365278028086, 1.556619009251110797930669149045, 2.40802978660195801609181929922, 3.92287286937819975286749595858, 4.89749191403005065278038565184, 5.6364884087664482965117211512, 6.72903822893589566561144105536, 7.55510064384400688789236899607, 8.10652654343894972839422993059, 9.56488289039469912231512921693, 10.76210970408102700477250516602, 11.82497130676574858909487664227, 12.675444261444892491129991292547, 13.083551774128536797112544140429, 13.86378440942964197330850868954, 14.741987843037323444186001595045, 15.837268206645999450528541768890, 16.87247310069277589464174001442, 17.32189290750549815746179212082, 18.01183199447036128385977870912, 19.69483540507833834790062241919, 20.13593418470432338729648116869, 20.73102399184596025447303281825, 22.11788573385607342960639231169, 22.59326439180523986026560760005

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