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Properties

Label 1-572-572.411-r0-0-0
Degree $1$
Conductor $572$
Sign $0.551 + 0.834i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (0.809 + 0.587i)3-s + (0.951 + 0.309i)5-s + (−0.587 − 0.809i)7-s + (0.309 + 0.951i)9-s + (0.587 + 0.809i)15-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s − i·21-s + 23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.309 − 0.951i)35-s + (0.587 + 0.809i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.951 + 0.309i)5-s + (−0.587 − 0.809i)7-s + (0.309 + 0.951i)9-s + (0.587 + 0.809i)15-s + (−0.309 + 0.951i)17-s + (−0.587 + 0.809i)19-s − i·21-s + 23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.309 − 0.951i)35-s + (0.587 + 0.809i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ 0.551 + 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.790475024 + 0.9624015190i\)
\(L(\frac12)\) \(\approx\) \(1.790475024 + 0.9624015190i\)
\(L(1)\) \(\approx\) \(1.457084321 + 0.4048231571i\)
\(L(1)\) \(\approx\) \(1.457084321 + 0.4048231571i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−23.10439111158340121288782278677, −22.21831289538362839406321452274, −21.23651292725900821026249309834, −20.75989036516354041882844001236, −19.64354848162326662701045347915, −19.03298826946302565970190778597, −18.11716413256626507971327391447, −17.51139065067596331244885302859, −16.34924582460281389726833987154, −15.3924824454759132731767955289, −14.60254043541760223579673314982, −13.587447948287535399597605809103, −13.058398214627609932867499519, −12.33220751208345397408713756194, −11.18094538520644718684290473404, −9.77637696528397877894420721019, −9.19694855784970777446253312925, −8.59106514489543393233497849769, −7.289602280995551610549358629915, −6.44421368278845298349898734946, −5.58127038135691886719476252536, −4.34381046906698418335066605463, −2.744857110911799059045419440254, −2.45564338366432470024642598538, −1.028476567187742919794844764205, 1.52421604319241288031197109224, 2.620927900760813669496197524216, 3.58744816447724165115669623171, 4.47879761551821593898104504833, 5.75334925244763795669137529762, 6.70917282793193473193659520111, 7.71814543894935227113837597953, 8.8277780593164466036240262798, 9.59716347954787028946630431537, 10.42626851798895633657199873032, 10.887110592527892402146853687796, 12.68361624313557406163540686837, 13.308213122587684892134102716822, 14.100298386795419913279344043797, 14.8208259383567736521685573648, 15.68596631310809375920575815421, 16.89743845812808824689591750916, 17.13887103137137416538227801339, 18.62278586635887037434978626851, 19.21861958700408856012944894863, 20.20422268795076505606954921942, 20.86976435026219120777334625018, 21.632737547542428146383199953638, 22.38344983673498328626946659888, 23.22432428717909266151410402756

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