Properties

Label 1-2011-2011.1341-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.609 - 0.792i$
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  + (0.720 − 0.693i)2-s + (0.999 − 0.00312i)3-s + (0.0390 − 0.999i)4-s + (−0.998 + 0.0593i)5-s + (0.718 − 0.695i)6-s + (0.805 + 0.592i)7-s + (−0.664 − 0.747i)8-s + (0.999 − 0.00625i)9-s + (−0.678 + 0.734i)10-s + (−0.995 + 0.0998i)11-s + (0.0359 − 0.999i)12-s + (0.974 + 0.223i)13-s + (0.991 − 0.130i)14-s + (−0.998 + 0.0624i)15-s + (−0.996 − 0.0780i)16-s + (0.467 − 0.884i)17-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)2-s + (0.999 − 0.00312i)3-s + (0.0390 − 0.999i)4-s + (−0.998 + 0.0593i)5-s + (0.718 − 0.695i)6-s + (0.805 + 0.592i)7-s + (−0.664 − 0.747i)8-s + (0.999 − 0.00625i)9-s + (−0.678 + 0.734i)10-s + (−0.995 + 0.0998i)11-s + (0.0359 − 0.999i)12-s + (0.974 + 0.223i)13-s + (0.991 − 0.130i)14-s + (−0.998 + 0.0624i)15-s + (−0.996 − 0.0780i)16-s + (0.467 − 0.884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (1341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.609 - 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.425919963 - 2.180190763i\)
\(L(\frac12)\) \(\approx\) \(4.425919963 - 2.180190763i\)
\(L(1)\) \(\approx\) \(1.939420424 - 0.7677572833i\)
\(L(1)\) \(\approx\) \(1.939420424 - 0.7677572833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.720 - 0.693i)T \)
3 \( 1 + (0.999 - 0.00312i)T \)
5 \( 1 + (-0.998 + 0.0593i)T \)
7 \( 1 + (0.805 + 0.592i)T \)
11 \( 1 + (-0.995 + 0.0998i)T \)
13 \( 1 + (0.974 + 0.223i)T \)
17 \( 1 + (0.467 - 0.884i)T \)
19 \( 1 + (-0.880 + 0.474i)T \)
23 \( 1 + (0.868 - 0.495i)T \)
29 \( 1 + (0.954 - 0.298i)T \)
31 \( 1 + (0.126 + 0.991i)T \)
37 \( 1 + (-0.540 - 0.841i)T \)
41 \( 1 + (0.447 + 0.894i)T \)
43 \( 1 + (0.379 + 0.925i)T \)
47 \( 1 + (0.422 + 0.906i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.218 + 0.975i)T \)
61 \( 1 + (-0.788 + 0.615i)T \)
67 \( 1 + (0.676 + 0.736i)T \)
71 \( 1 + (-0.929 - 0.369i)T \)
73 \( 1 + (-0.701 - 0.713i)T \)
79 \( 1 + (-0.151 + 0.988i)T \)
83 \( 1 + (-0.233 + 0.972i)T \)
89 \( 1 + (0.524 - 0.851i)T \)
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

−20.18195765178546581781408668558, −19.06980948791486157402199191542, −18.58086733611680619404810050197, −17.49098849209773589124581270234, −16.84745803816035072192527425759, −15.70870423464385538491994256077, −15.521663459660445914109286580456, −14.83600490032652793615996658634, −14.073029522699363798756886967140, −13.34190794300449794462764926681, −12.84327150799604625230150540546, −11.96050058128405721304399264639, −10.9232427497996900711389528231, −10.45957344210093233404829972929, −8.81359535970262690565265492917, −8.46237163496872192002540373503, −7.74135648686691478089407860714, −7.318050963110449241457322363525, −6.31177196982843770207934744301, −5.11720227299702601643889368959, −4.44311666491113211913001464022, −3.71573964671274321872318319632, −3.12708157112690115113227390602, −2.03363766599336778054424767875, −0.7393845702924940240705352649, 0.814136770037116884041866258, 1.716563840045024385565323294838, 2.809823950274316958997194653592, 3.106484303671908221719649097638, 4.375141670060289410617470965572, 4.62304044313063693093905687218, 5.742120643420258666137402019679, 6.852960251738478904015560191718, 7.73786361413608896190363518624, 8.50087318721909331842378527761, 9.036673795155535558793056892115, 10.21886149735277843149515201089, 10.85639964702425366660438854180, 11.55064310850374448968516825144, 12.471957780120334672310779801534, 12.86819061302436209394892881571, 13.91680748485492356012134568957, 14.452474138404587968434165746640, 15.10028460999141342329627469702, 15.76779941871530531828592731467, 16.22252729596439624779156703352, 18.05706731715578034979115648739, 18.39504674283048810331808243523, 19.19154205642374515763393183851, 19.58594051463094601026490764181

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