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Properties

Label 1-2011-2011.69-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.687 + 0.725i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (−0.616 + 0.787i)2-s + (0.866 + 0.498i)3-s + (−0.239 − 0.970i)4-s + (0.880 + 0.474i)5-s + (−0.926 + 0.375i)6-s + (0.666 + 0.745i)7-s + (0.912 + 0.409i)8-s + (0.502 + 0.864i)9-s + (−0.916 + 0.401i)10-s + (0.542 + 0.839i)11-s + (0.276 − 0.961i)12-s + (0.993 + 0.112i)13-s + (−0.997 + 0.0655i)14-s + (0.526 + 0.849i)15-s + (−0.884 + 0.465i)16-s + (0.483 − 0.875i)17-s + ⋯
L(s)  = 1  + (−0.616 + 0.787i)2-s + (0.866 + 0.498i)3-s + (−0.239 − 0.970i)4-s + (0.880 + 0.474i)5-s + (−0.926 + 0.375i)6-s + (0.666 + 0.745i)7-s + (0.912 + 0.409i)8-s + (0.502 + 0.864i)9-s + (−0.916 + 0.401i)10-s + (0.542 + 0.839i)11-s + (0.276 − 0.961i)12-s + (0.993 + 0.112i)13-s + (−0.997 + 0.0655i)14-s + (0.526 + 0.849i)15-s + (−0.884 + 0.465i)16-s + (0.483 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.687 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.687 + 0.725i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.687 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504347936 + 3.498148643i\)
\(L(\frac12)\) \(\approx\) \(1.504347936 + 3.498148643i\)
\(L(1)\) \(\approx\) \(1.184224568 + 0.9961701349i\)
\(L(1)\) \(\approx\) \(1.184224568 + 0.9961701349i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.616 + 0.787i)T \)
3 \( 1 + (0.866 + 0.498i)T \)
5 \( 1 + (0.880 + 0.474i)T \)
7 \( 1 + (0.666 + 0.745i)T \)
11 \( 1 + (0.542 + 0.839i)T \)
13 \( 1 + (0.993 + 0.112i)T \)
17 \( 1 + (0.483 - 0.875i)T \)
19 \( 1 + (-0.696 - 0.717i)T \)
23 \( 1 + (0.261 - 0.965i)T \)
29 \( 1 + (0.931 + 0.363i)T \)
31 \( 1 + (-0.750 - 0.660i)T \)
37 \( 1 + (0.854 - 0.520i)T \)
41 \( 1 + (-0.850 - 0.525i)T \)
43 \( 1 + (-0.830 - 0.557i)T \)
47 \( 1 + (0.537 - 0.843i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.780 + 0.625i)T \)
61 \( 1 + (0.754 + 0.656i)T \)
67 \( 1 + (0.994 - 0.109i)T \)
71 \( 1 + (-0.944 + 0.328i)T \)
73 \( 1 + (-0.795 + 0.605i)T \)
79 \( 1 + (0.184 + 0.982i)T \)
83 \( 1 + (0.989 - 0.143i)T \)
89 \( 1 + (0.858 + 0.512i)T \)
97 \( 1 + (0.553 + 0.833i)T \)
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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

−19.488283023525719763916087903168, −18.88382053021287977861341050979, −18.09280575779156497087836956561, −17.50082949482398246233054231658, −16.83763953587135039916613990686, −16.15617262626638088632927476351, −14.81927496987225131448280262100, −14.085511847777400641263937512240, −13.46370438253351759350310913606, −13.030430324619753978049371357772, −12.13635230822879718991244062787, −11.26237692879262265159221980289, −10.44259034307426595452709063864, −9.80269706867662120382409747425, −8.898681688807003991166937036564, −8.3374214957121414490221534749, −7.90409838732219623698006618322, −6.71782120495603424291278292417, −5.970608776551224631846674026047, −4.61198729627446463600901527176, −3.70795831063249237999663092495, −3.16026358176256239569602848630, −1.80940071860012686758381615645, −1.4477013837925846585782856292, −0.772479052379452950833782546872, 1.10104735600930152801746499523, 2.06341635418672402813512926822, 2.59215611154155829882775855254, 4.01357313289435547588769747915, 4.90207096817805380831450865527, 5.5663887625756830588596978435, 6.60141388313767955224004019245, 7.18959558069853948103781898169, 8.20655769625214004379578718718, 8.99087594717114370792112635331, 9.211097730883709344057762086606, 10.241972476306598110289049383097, 10.71540998209164830315979873963, 11.71966853803510904147722960914, 13.04144718097696922038105191607, 13.76814504907436568306486975307, 14.45362372131243592682606448376, 14.91859443633446878410800466192, 15.47692297209064691568933862937, 16.39455244122191147816694179331, 17.09192334769721433380885519384, 17.99386837153977881356764492973, 18.46473712209200183032794177561, 18.99761271640807358635324355556, 20.14547325471031164627422122464

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