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Properties

Label 1-2011-2011.86-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.865 + 0.500i$
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  + (0.239 − 0.970i)2-s + (0.914 − 0.403i)3-s + (−0.884 − 0.465i)4-s + (0.0453 + 0.998i)5-s + (−0.172 − 0.984i)6-s + (−0.910 + 0.412i)7-s + (−0.664 + 0.747i)8-s + (0.673 − 0.738i)9-s + (0.980 + 0.195i)10-s + (−0.739 + 0.672i)11-s + (−0.997 − 0.0687i)12-s + (0.0889 − 0.996i)13-s + (0.181 + 0.983i)14-s + (0.444 + 0.895i)15-s + (0.566 + 0.824i)16-s + (0.969 + 0.244i)17-s + ⋯
L(s)  = 1  + (0.239 − 0.970i)2-s + (0.914 − 0.403i)3-s + (−0.884 − 0.465i)4-s + (0.0453 + 0.998i)5-s + (−0.172 − 0.984i)6-s + (−0.910 + 0.412i)7-s + (−0.664 + 0.747i)8-s + (0.673 − 0.738i)9-s + (0.980 + 0.195i)10-s + (−0.739 + 0.672i)11-s + (−0.997 − 0.0687i)12-s + (0.0889 − 0.996i)13-s + (0.181 + 0.983i)14-s + (0.444 + 0.895i)15-s + (0.566 + 0.824i)16-s + (0.969 + 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.813775023 + 0.4868584818i\)
\(L(\frac12)\) \(\approx\) \(1.813775023 + 0.4868584818i\)
\(L(1)\) \(\approx\) \(1.179342276 - 0.4470104490i\)
\(L(1)\) \(\approx\) \(1.179342276 - 0.4470104490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.239 - 0.970i)T \)
3 \( 1 + (0.914 - 0.403i)T \)
5 \( 1 + (0.0453 + 0.998i)T \)
7 \( 1 + (-0.910 + 0.412i)T \)
11 \( 1 + (-0.739 + 0.672i)T \)
13 \( 1 + (0.0889 - 0.996i)T \)
17 \( 1 + (0.969 + 0.244i)T \)
19 \( 1 + (0.959 + 0.280i)T \)
23 \( 1 + (0.995 + 0.0998i)T \)
29 \( 1 + (-0.871 - 0.490i)T \)
31 \( 1 + (-0.904 - 0.426i)T \)
37 \( 1 + (-0.458 - 0.888i)T \)
41 \( 1 + (-0.887 + 0.460i)T \)
43 \( 1 + (0.997 + 0.0749i)T \)
47 \( 1 + (0.992 + 0.121i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.860 - 0.509i)T \)
61 \( 1 + (-0.984 + 0.174i)T \)
67 \( 1 + (-0.976 - 0.217i)T \)
71 \( 1 + (-0.999 - 0.0406i)T \)
73 \( 1 + (-0.350 + 0.936i)T \)
79 \( 1 + (-0.967 + 0.253i)T \)
83 \( 1 + (0.0265 + 0.999i)T \)
89 \( 1 + (-0.901 + 0.432i)T \)
97 \( 1 + (0.388 + 0.921i)T \)
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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

−19.65821022891695387469577046804, −18.90771969297447991752252226439, −18.40788355889962341210506138838, −17.04097354759880231823007134904, −16.52317901633673464123539501141, −16.11019040344710037981027458864, −15.52041556350351279172011583568, −14.542382801107483039727263975841, −13.7741361933783202082019879655, −13.37049944105409407331746622069, −12.754534459100881887037236416775, −11.822868452965312967429420562184, −10.450178615139312328576434246858, −9.68050456932866532042081238484, −8.93728215482826841367484568486, −8.68593034111249039014392368287, −7.39004167194845603456827195373, −7.25651711357812229494584689766, −5.82911105309644602788572982692, −5.20391412170272280505530924540, −4.396540522250231290745576916136, −3.51242575558470242294157783864, −2.99183397135761702117705876796, −1.41476128470003255216017609193, −0.299868736612445210339513235142, 0.925944716701341740465622301151, 2.07685404804745540144066698925, 2.77488643206458846904905624918, 3.285130349627370882345234041470, 3.92836433960332967743134374858, 5.42649591699663711910028797206, 5.9349622069370917449612593613, 7.27652384045858488263490209900, 7.65358797826978098945691094788, 8.81214645522671840885295295676, 9.61739829446328523005041584406, 10.09049475792565310638145830823, 10.77938972546537699960488815600, 11.878305415737563973612956193795, 12.60977373964060023611802696308, 13.10202048232472189975657700652, 13.78307015619385611137288631510, 14.68714036169661377790395483626, 15.12108298048678190851511419630, 15.80057674039473215617262044492, 17.231825276368981115821174150456, 18.242587357708532059161672958365, 18.46811669853477530805087956212, 19.126357032610997761238053532, 19.77439505811136378721309508078

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