Properties

Label 1-2013-2013.1889-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

Related objects

Downloads

Learn more

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: $\chi_{2013} (1889, \cdot )$
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 \( 1 + (0.978 + 0.207i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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