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Properties

Label 1-211-211.11-r0-0-0
Degree $1$
Conductor $211$
Sign $0.911 - 0.411i$
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  + (0.134 − 0.990i)2-s + (0.473 + 0.880i)3-s + (−0.963 − 0.266i)4-s + (0.473 − 0.880i)5-s + (0.936 − 0.351i)6-s + (0.134 + 0.990i)7-s + (−0.393 + 0.919i)8-s + (−0.550 + 0.834i)9-s + (−0.809 − 0.587i)10-s + (0.983 − 0.178i)11-s + (−0.222 − 0.974i)12-s + (0.858 + 0.512i)13-s + 14-s + 15-s + (0.858 + 0.512i)16-s + (−0.995 − 0.0896i)17-s + ⋯
L(s)  = 1  + (0.134 − 0.990i)2-s + (0.473 + 0.880i)3-s + (−0.963 − 0.266i)4-s + (0.473 − 0.880i)5-s + (0.936 − 0.351i)6-s + (0.134 + 0.990i)7-s + (−0.393 + 0.919i)8-s + (−0.550 + 0.834i)9-s + (−0.809 − 0.587i)10-s + (0.983 − 0.178i)11-s + (−0.222 − 0.974i)12-s + (0.858 + 0.512i)13-s + 14-s + 15-s + (0.858 + 0.512i)16-s + (−0.995 − 0.0896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448892563 - 0.3116884600i\)
\(L(\frac12)\) \(\approx\) \(1.448892563 - 0.3116884600i\)
\(L(1)\) \(\approx\) \(1.278373065 - 0.2820858509i\)
\(L(1)\) \(\approx\) \(1.278373065 - 0.2820858509i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.134 - 0.990i)T \)
3 \( 1 + (0.473 + 0.880i)T \)
5 \( 1 + (0.473 - 0.880i)T \)
7 \( 1 + (0.134 + 0.990i)T \)
11 \( 1 + (0.983 - 0.178i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (-0.995 - 0.0896i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (0.983 + 0.178i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.550 - 0.834i)T \)
53 \( 1 + (-0.963 + 0.266i)T \)
59 \( 1 + (0.753 + 0.657i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.393 - 0.919i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.691 + 0.722i)T \)
97 \( 1 + (-0.0448 - 0.998i)T \)
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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

−26.606472174480126790703521853758, −25.563185380170900157967556759530, −25.0312732674324200175708958915, −24.10127376224798137366239571934, −22.940621910641754055897981107180, −22.66730619225002360953771412430, −21.16400276250181140661440943008, −19.9703879720523006088684675528, −18.940238334736858554243954692612, −17.880767777192882550226945742059, −17.52432875027031818531114119581, −16.24927032388143854259223830421, −14.84608537764491528464082004659, −14.23708993486386770982539725759, −13.55585707757020081377453896175, −12.60350782268190047609994767803, −11.059079062923501792735312948683, −9.76951195743424487098662645572, −8.58212593062400551974092217154, −7.58423151797976430190455029936, −6.632419857892048821021450788027, −6.12379763860119303464526482354, −4.226122250934841740437965621997, −3.12326359944537537216123682813, −1.296773873169782677705000841090, 1.56083949687387329524496460234, 2.738348046577993297794553849157, 4.0743078621506883138869863753, 4.93455691885301668013367854428, 6.01253586568078736693008463688, 8.47258201122760826515426819150, 9.07942364446563853860812854497, 9.585449109042167838817178814484, 11.126328311789137695767378888491, 11.75437166066759299135661214431, 13.15236855314579299046114662346, 13.83417057802931778626090209696, 14.97411455035130513114165294357, 16.00218582958677222898199725110, 17.17817313351221426449822159102, 18.17719867480086023054850766443, 19.47439636871186281952290575616, 20.09514292364292218897484745902, 21.08656821214924302109618108540, 21.72827983166727523478908868613, 22.281039700289550645711182983581, 23.765644207561650060132926149770, 24.858963824076744120620351352403, 25.75357263506643073049272856747, 26.91517008861094048232464641899

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