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Properties

Label 1-2013-2013.1250-r1-0-0
Degree $1$
Conductor $2013$
Sign $-0.0919 + 0.995i$
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  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (−0.866 − 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.743 − 0.669i)10-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.994 − 0.104i)17-s + (−0.5 − 0.866i)19-s + (−0.309 + 0.951i)20-s + (0.951 + 0.309i)23-s + (0.669 − 0.743i)25-s + (−0.207 − 0.978i)26-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.913 − 0.406i)5-s + (−0.866 − 0.5i)7-s + (0.951 + 0.309i)8-s + (−0.743 − 0.669i)10-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (−0.994 − 0.104i)17-s + (−0.5 − 0.866i)19-s + (−0.309 + 0.951i)20-s + (0.951 + 0.309i)23-s + (0.669 − 0.743i)25-s + (−0.207 − 0.978i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0919 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0919 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.0919 + 0.995i$
Analytic conductor: \(216.326\)
Root analytic conductor: \(216.326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (1250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (1:\ ),\ -0.0919 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1055553388 - 0.1157539443i\)
\(L(\frac12)\) \(\approx\) \(-0.1055553388 - 0.1157539443i\)
\(L(1)\) \(\approx\) \(0.6719661740 - 0.4120544469i\)
\(L(1)\) \(\approx\) \(0.6719661740 - 0.4120544469i\)

Euler product

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

−20.168418248527604975939847122777, −19.18727597773370440213720429184, −18.71889899024919973910001214082, −18.05705894419029164884451516658, −17.401053112291592506566021544299, −16.64260407136911332599372471462, −15.96296175989997838409118366461, −15.237597419635134762460324510713, −14.62879282204183407167302946510, −13.69290327344489808629928034711, −13.20464461809262999681799600929, −12.49942507075642371659514723759, −11.045405903795080964874447006, −10.51291247451047312321347022182, −9.70108399417171294494957580724, −8.99348923711503132932485366364, −8.5036213447506569334698403873, −7.3283978870753054369092486919, −6.52125497348017924791619287168, −6.07367221879633729721581849822, −5.43768527682935110157064589909, −4.31513949380897939156839671220, −3.28114124955795867369324416048, −2.21212885138880803349611932348, −1.25561981822408813340175039713, 0.03521170910676018452292531710, 0.96182707177950749884577571094, 1.83357327005757977854881637467, 2.73328652274440314155483343874, 3.585252170722473820807598525328, 4.45588417469022431519071732673, 5.30165818276406174014115289804, 6.46160648784055961408741276840, 7.018962870061481494582318222457, 8.29352067097844797881324524779, 9.07783686822450352399196747654, 9.39095404176292880395358738360, 10.3725530146787199360097633523, 10.90913299109827438562997030590, 11.69189071930830722894230607183, 12.826771076754343122505420439005, 13.3793773117043386408949282208, 13.44111202740092549971396790034, 14.69074456935832898214106131754, 15.83790166839133332493061357667, 16.54871243949362378961938812708, 17.19665927357488305027683641619, 17.7788973655072789202463204756, 18.59105914506779689717080628368, 19.24184724425442977565377443362

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