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Properties

Label 1-2013-2013.224-r1-0-0
Degree $1$
Conductor $2013$
Sign $0.0586 - 0.998i$
Analytic cond. $216.326$
Root an. cond. $216.326$
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)2-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s − 7-s + (0.309 − 0.951i)8-s + (−0.309 − 0.951i)10-s + (0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.809 − 0.587i)17-s + 19-s + (−0.309 + 0.951i)20-s + (0.309 − 0.951i)23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 + 0.587i)5-s − 7-s + (0.309 − 0.951i)8-s + (−0.309 − 0.951i)10-s + (0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (−0.809 − 0.587i)17-s + 19-s + (−0.309 + 0.951i)20-s + (0.309 − 0.951i)23-s + (0.309 + 0.951i)25-s + (0.309 − 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0586 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0586 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $0.0586 - 0.998i$
Analytic conductor: \(216.326\)
Root analytic conductor: \(216.326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (1:\ ),\ 0.0586 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6190588551 - 0.5837601279i\)
\(L(\frac12)\) \(\approx\) \(0.6190588551 - 0.5837601279i\)
\(L(1)\) \(\approx\) \(0.7031942523 - 0.07850786892i\)
\(L(1)\) \(\approx\) \(0.7031942523 - 0.07850786892i\)

Euler product

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

−19.91472806975956828796404979825, −19.15864924940034270201053512425, −18.2545238281218060702213338441, −17.75176267998492532837052951008, −16.96013271321563689906797291349, −16.49911744742829024278085298639, −15.53308708507467993851770442184, −15.25495634292653033505109521140, −14.0197664672884760202658986347, −13.319187029914975388471477241256, −12.85817273911122732345217633652, −11.66158122262979374481007045895, −10.79083121092026859062033730670, −9.91171666507289704428382161735, −9.53988983362893145418278967606, −8.79645934665322485649823688692, −7.98104399619697711048178176649, −7.125587631722335707143839096611, −6.21523731726547697647107604390, −5.72296184207521439125375589009, −4.98706774743435328732791645848, −3.6775569084212173194967340589, −2.59871387113760729522257482855, −1.61337976588722414881567685270, −0.72311455811109529543817203807, 0.25523182130611689628675133614, 1.48586847776781500098023563213, 2.29698746242243560175244113712, 3.10323890629702034503385752, 3.7804597339886305325054661183, 5.03868335800670859310974134448, 6.18701035045274140942148742963, 6.93979558099919048927817944545, 7.2741492822819087481295583727, 8.93530914294681977620816784355, 8.99454335553937033405490926283, 9.92012613914837580203112598300, 10.57679522419241915261015142733, 11.22854930964832479898150885186, 12.13950086443786036811726605878, 12.85835522741104171378548546703, 13.68325774751593606510116183449, 14.18236188120850566218303591892, 15.505584667608807012828328469076, 16.10139387828532983418113998460, 16.80438823922406295901768967157, 17.521331839784366319772936371673, 18.33841048487704982665884458234, 18.76535431642611029794050929742, 19.388192120193530094941146011939

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