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Properties

Label 1-211-211.79-r0-0-0
Degree $1$
Conductor $211$
Sign $0.577 - 0.816i$
Analytic cond. $0.979879$
Root an. cond. $0.979879$
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  + (−0.995 + 0.0896i)2-s + (0.753 − 0.657i)3-s + (0.983 − 0.178i)4-s + (0.753 + 0.657i)5-s + (−0.691 + 0.722i)6-s + (−0.995 − 0.0896i)7-s + (−0.963 + 0.266i)8-s + (0.134 − 0.990i)9-s + (−0.809 − 0.587i)10-s + (−0.393 − 0.919i)11-s + (0.623 − 0.781i)12-s + (0.936 − 0.351i)13-s + 14-s + 15-s + (0.936 − 0.351i)16-s + (−0.550 − 0.834i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (0.753 − 0.657i)3-s + (0.983 − 0.178i)4-s + (0.753 + 0.657i)5-s + (−0.691 + 0.722i)6-s + (−0.995 − 0.0896i)7-s + (−0.963 + 0.266i)8-s + (0.134 − 0.990i)9-s + (−0.809 − 0.587i)10-s + (−0.393 − 0.919i)11-s + (0.623 − 0.781i)12-s + (0.936 − 0.351i)13-s + 14-s + 15-s + (0.936 − 0.351i)16-s + (−0.550 − 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8870590873 - 0.4593297897i\)
\(L(\frac12)\) \(\approx\) \(0.8870590873 - 0.4593297897i\)
\(L(1)\) \(\approx\) \(0.8935649370 - 0.2213931033i\)
\(L(1)\) \(\approx\) \(0.8935649370 - 0.2213931033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
5 \( 1 + (0.753 + 0.657i)T \)
7 \( 1 + (-0.995 - 0.0896i)T \)
11 \( 1 + (-0.393 - 0.919i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (-0.550 - 0.834i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.936 - 0.351i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
41 \( 1 + (-0.0448 + 0.998i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.134 + 0.990i)T \)
53 \( 1 + (0.983 + 0.178i)T \)
59 \( 1 + (-0.0448 + 0.998i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.963 - 0.266i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.858 + 0.512i)T \)
97 \( 1 + (0.473 + 0.880i)T \)
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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

−26.57132237829573154817880538767, −25.96721847891764276449206547024, −25.30133378114769920090605840659, −24.59373013906923490098035432691, −23.03280773445925509847883152235, −21.681958727730369015290401435291, −20.92521757312123661130114006312, −20.229829982077020893281598413152, −19.37049480288407904779831053013, −18.34163731623517423659369851976, −17.24664675385810083611207786216, −16.193284823865029503114229794263, −15.76561084041816533282977700035, −14.45016988467714751875235215834, −13.15794924005772334098994912452, −12.31902151364179704167976107292, −10.47576222884444498303398555566, −10.079582590087012026583842651849, −8.96304970398519519252032462119, −8.47353206476460535954068037538, −6.94974791284738108264707420006, −5.773648571748335877699583746129, −4.13882632239857524051193039903, −2.76161539292939035508738984292, −1.65828537837773371233383017454, 1.01181640813167190585994206080, 2.66080946617083872115037118219, 3.14977888805212180147923004516, 5.931065740270596890996024561606, 6.61721489103274052067688037101, 7.59197896193766256287886323590, 8.82986007944245106753111171339, 9.53205557976842276771408355317, 10.608022100132435248161458539757, 11.705352231204035293987246915593, 13.37069975143173323488088053514, 13.67424056150852707063043519097, 15.26846305844758785224209227615, 15.92667687398444739045717320147, 17.330597323134286458569186373033, 18.19149047229384984620101255684, 18.859744330101323847037555232691, 19.62758371496674345153207801392, 20.63256224701293263615738010870, 21.575950198900380030475091905504, 22.93225216349950637643953670992, 24.10563354961518732157588536925, 25.04135697975775363702328945906, 25.77891277655316955149117035085, 26.26496087888374334243704848170

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