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Properties

Label 1-571-571.66-r0-0-0
Degree $1$
Conductor $571$
Sign $-0.548 - 0.835i$
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  + (−0.298 + 0.954i)2-s + (−0.899 + 0.436i)3-s + (−0.821 − 0.569i)4-s + (−0.609 − 0.792i)5-s + (−0.148 − 0.988i)6-s + (0.746 − 0.665i)7-s + (0.789 − 0.614i)8-s + (0.618 − 0.785i)9-s + (0.938 − 0.345i)10-s + (−0.170 − 0.985i)11-s + (0.988 + 0.153i)12-s + (−0.421 − 0.906i)13-s + (0.411 + 0.911i)14-s + (0.894 + 0.446i)15-s + (0.350 + 0.936i)16-s + (0.266 − 0.963i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)2-s + (−0.899 + 0.436i)3-s + (−0.821 − 0.569i)4-s + (−0.609 − 0.792i)5-s + (−0.148 − 0.988i)6-s + (0.746 − 0.665i)7-s + (0.789 − 0.614i)8-s + (0.618 − 0.785i)9-s + (0.938 − 0.345i)10-s + (−0.170 − 0.985i)11-s + (0.988 + 0.153i)12-s + (−0.421 − 0.906i)13-s + (0.411 + 0.911i)14-s + (0.894 + 0.446i)15-s + (0.350 + 0.936i)16-s + (0.266 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(571\)
Sign: $-0.548 - 0.835i$
Analytic conductor: \(2.65171\)
Root analytic conductor: \(2.65171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{571} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 571,\ (0:\ ),\ -0.548 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1519781027 - 0.2816209459i\)
\(L(\frac12)\) \(\approx\) \(0.1519781027 - 0.2816209459i\)
\(L(1)\) \(\approx\) \(0.5196331439 + 0.02370975386i\)
\(L(1)\) \(\approx\) \(0.5196331439 + 0.02370975386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad571 \( 1 \)
good2 \( 1 + (-0.298 + 0.954i)T \)
3 \( 1 + (-0.899 + 0.436i)T \)
5 \( 1 + (-0.609 - 0.792i)T \)
7 \( 1 + (0.746 - 0.665i)T \)
11 \( 1 + (-0.170 - 0.985i)T \)
13 \( 1 + (-0.421 - 0.906i)T \)
17 \( 1 + (0.266 - 0.963i)T \)
19 \( 1 + (-0.693 - 0.720i)T \)
23 \( 1 + (-0.340 + 0.940i)T \)
29 \( 1 + (-0.754 - 0.656i)T \)
31 \( 1 + (-0.986 - 0.164i)T \)
37 \( 1 + (0.731 + 0.681i)T \)
41 \( 1 + (0.0275 + 0.999i)T \)
43 \( 1 + (-0.556 + 0.831i)T \)
47 \( 1 + (0.635 - 0.771i)T \)
53 \( 1 + (-0.319 + 0.947i)T \)
59 \( 1 + (-0.401 - 0.915i)T \)
61 \( 1 + (0.509 + 0.860i)T \)
67 \( 1 + (-0.660 - 0.750i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.224 - 0.974i)T \)
79 \( 1 + (-0.126 + 0.991i)T \)
83 \( 1 + (0.930 + 0.366i)T \)
89 \( 1 + (-0.782 + 0.622i)T \)
97 \( 1 + (0.999 - 0.0220i)T \)
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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

−23.50454718316602667720227841812, −22.46574764683049474442854317881, −21.97193359005258771752165754221, −21.17073092135072710224149787332, −20.12343552126021829881172810864, −19.009832114764444438164729502283, −18.69180264015187907816582313597, −17.90888928090303443920186708844, −17.16098116846566845450535129109, −16.2288843578781352494316373532, −14.84328967354002993183752473992, −14.325161406841403835421544023106, −12.74562217436242627878560217742, −12.31124841101650100534707944642, −11.56796713016983184767869676158, −10.769770976647958014784060519620, −10.17312577555952088583886675422, −8.83313521239074137078368873949, −7.80648896943126515251848919714, −7.100635062285820826960606504757, −5.82164158641886565906530969093, −4.64790392576227719796177441477, −3.91156851693223452907395388007, −2.23898260966605454241131437694, −1.77138287070982308224518860532, 0.24045490649336010036067423673, 1.149903907839093850372490220340, 3.60838098829436217559300279301, 4.62428659625434082939207336391, 5.19227079187390552510079056386, 6.05538936874818137269859488585, 7.39002567283255792564035276898, 7.9175437919020577650268167644, 9.015908105601495200759188000247, 9.90909376902338613030309275245, 10.97839238593883696198893917206, 11.59501484484478262877002221715, 12.91287673193192014703381603626, 13.624711144355504731545519117056, 14.94155629021630095452699013979, 15.483904488185383829500903923360, 16.52788511357808805944706141175, 16.79996758591329214833921718037, 17.70276659871741175767540623388, 18.43462568522734501198244410982, 19.606108031313645932598019495328, 20.44343312084204124741619695362, 21.47509512666780588301080081216, 22.33149845649287202929301072309, 23.38874887772558487013545845298

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