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Properties

Label 1-2013-2013.59-r0-0-0
Degree $1$
Conductor $2013$
Sign $0.175 - 0.984i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.5 − 0.866i)5-s + (−0.207 − 0.978i)7-s + (−0.587 − 0.809i)8-s + (0.743 − 0.669i)10-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.866 − 0.5i)17-s + (0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + (0.951 − 0.309i)23-s + (−0.5 + 0.866i)25-s + (0.866 + 0.5i)26-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (−0.5 − 0.866i)5-s + (−0.207 − 0.978i)7-s + (−0.587 − 0.809i)8-s + (0.743 − 0.669i)10-s + (0.669 − 0.743i)13-s + (0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (−0.866 − 0.5i)17-s + (0.978 + 0.207i)19-s + (0.809 + 0.587i)20-s + (0.951 − 0.309i)23-s + (−0.5 + 0.866i)25-s + (0.866 + 0.5i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ 0.175 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8046656301 - 0.6739848851i\)
\(L(\frac12)\) \(\approx\) \(0.8046656301 - 0.6739848851i\)
\(L(1)\) \(\approx\) \(0.9212593940 + 0.04580803454i\)
\(L(1)\) \(\approx\) \(0.9212593940 + 0.04580803454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.951 - 0.309i)T \)
29 \( 1 + (0.994 - 0.104i)T \)
31 \( 1 + (-0.406 - 0.913i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + T \)
43 \( 1 + (0.207 + 0.978i)T \)
47 \( 1 + (-0.913 + 0.406i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.406 + 0.913i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−19.94592397405870963666092925552, −19.27965303441354233611309949735, −18.90904097375155046395031585364, −18.01139501598550969415512500417, −17.72542659010635364579183884774, −16.297103013499153883865705369999, −15.51893499315805906844232801276, −14.97639197861135027208372149189, −14.07392185183435420214227907547, −13.52487176767394211987465941069, −12.520805770985156403855908673384, −11.91454630765062372601560401588, −11.23992024210559225530041009140, −10.70855850757186385097724198265, −9.791215370067492984911073177443, −8.88950329916315737994476170156, −8.502899862628342751964691937323, −7.18198677314471876577398847495, −6.40693255225003866891905778836, −5.52313920611043453005313135740, −4.62105514450781373065440275920, −3.6769363246472825819315884501, −3.023438444959677743473302064043, −2.2743062032279423816200475981, −1.27017063835037623574953429352, 0.40014671392929750081014708076, 1.1845333044433355746951842415, 3.023572747255698770800578618973, 3.79403540029082670652335525351, 4.57054711427592575728267778086, 5.19057233957555298888964178148, 6.1488275439316872958321411898, 7.021452128805325984768756296882, 7.69230244522639955435477523684, 8.34933855783786777913717033517, 9.14317587397463773823640570945, 9.85539048346189201908614636239, 10.92561137016536162775378273961, 11.74952388029018985178798746710, 12.8774833088411356147514006556, 13.11139795942823327459267126514, 13.90326241373634630973121022728, 14.70780678193037450313579832074, 15.705022260494005866506948235626, 16.053023572492413622345912685199, 16.66210224318445589105915509237, 17.5645776886623178926338217846, 17.93631063219371328292098229053, 19.07096164717654745717591738694, 19.77816001577565337684337286466

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