Properties

Label 1-571-571.387-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.763 - 0.645i$
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 + (0.309 − 0.951i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + 8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + 16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + 2-s + (0.309 − 0.951i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 − 0.951i)7-s + 8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + 16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7204616123 - 1.969005154i\)
\(L(\frac12)\) \(\approx\) \(0.7204616123 - 1.969005154i\)
\(L(1)\) \(\approx\) \(1.368855669 - 0.9709798477i\)
\(L(1)\) \(\approx\) \(1.368855669 - 0.9709798477i\)

Euler product

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

−23.45114796562153578258052988016, −22.53215199158199507200594629608, −21.92863253873550180137699024541, −21.24933145316889284268711389668, −20.606051099528126008799889576721, −19.32274346251424402818890713624, −19.126267327340831720312902828802, −17.57695034617819062364500614905, −16.360652075675009679023823428475, −15.60361519679566240449518429326, −15.27794532891696020724898819668, −14.37248631473510799433449869064, −13.69398225897739670186618392372, −12.33918307433721620535319896240, −11.5932558436248897546817239752, −10.932851274537442735317937206220, −10.04369910915846221207845529982, −8.69884486126357371262276457599, −7.90478141536281945175013598596, −6.78058715265252167458951386468, −5.617153210346486434991461254133, −4.79139393471479661977361371865, −3.98691762300402416911207677332, −2.84395997906474959027632216362, −2.37331779961891567577857212530, 0.6986100155355982337596936362, 2.08726499001818929049024874498, 3.009303045086737281089708658828, 4.35355798181221805687727105968, 4.82947208441376306609313376397, 6.27031510897920005398344335568, 7.2182370416930783461870024918, 7.76728194610666947763782155270, 8.631198460788824261158483091739, 10.36881814985495641283759443627, 11.11293219748024950318009205106, 12.24134645455000764633348646514, 12.75669825841396362631958406828, 13.35716724347570693335422711328, 14.409609324043726445527077808682, 15.127217948619415347373703684466, 15.95142089350710006945577124104, 17.170925281632166279948168981080, 17.65031417942528504863003781884, 19.29592918585957536289516967177, 19.63822804405652768125967129139, 20.48036101283860086424562813786, 20.98423451667934742970674830910, 22.40084414305921031926863461908, 23.27734173263218106475711094698

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