Properties

Degree 1
Conductor 2011
Sign $0.609 + 0.792i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(2011\)
\( \varepsilon \)  =  $0.609 + 0.792i$
motivic weight  =  \(0\)
character  :  $\chi_{2011} (3, \cdot )$
Sato-Tate  :  $\mu(2010)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 2011,\ (1:\ ),\ 0.609 + 0.792i)$
$L(\chi,\frac{1}{2})$  $\approx$  $4.425919963 + 2.180190763i$
$L(\frac12,\chi)$  $\approx$  $4.425919963 + 2.180190763i$
$L(\chi,1)$  $\approx$  1.939420424 + 0.7677572833i
$L(1,\chi)$  $\approx$  1.939420424 + 0.7677572833i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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