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Properties

Label 1-211-211.148-r0-0-0
Degree $1$
Conductor $211$
Sign $0.149 - 0.988i$
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  + (−0.900 − 0.433i)2-s + (−0.900 − 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)6-s + (−0.900 + 0.433i)7-s + (−0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + 10-s + (−0.900 − 0.433i)11-s + (−0.222 − 0.974i)12-s + (−0.222 + 0.974i)13-s + 14-s + 15-s + (−0.222 + 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.900 − 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)6-s + (−0.900 + 0.433i)7-s + (−0.222 − 0.974i)8-s + (0.623 + 0.781i)9-s + 10-s + (−0.900 − 0.433i)11-s + (−0.222 − 0.974i)12-s + (−0.222 + 0.974i)13-s + 14-s + 15-s + (−0.222 + 0.974i)16-s + (−0.222 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.149 - 0.988i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.149 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2187454766 - 0.1881358824i\)
\(L(\frac12)\) \(\approx\) \(0.2187454766 - 0.1881358824i\)
\(L(1)\) \(\approx\) \(0.3782492314 - 0.09329875417i\)
\(L(1)\) \(\approx\) \(0.3782492314 - 0.09329875417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−26.79969649645730375456357588213, −26.39848570517977423409951231824, −25.06455596861993638518049713759, −24.01296778236233334238423027650, −23.19071927474848247014516328883, −22.65811265103990123391831375650, −21.018594710222461366499096311722, −20.09899271505094066147873369077, −19.2888448780554095564236483092, −18.14576708224377418669393144406, −17.24267001093341730141856527314, −16.39961168361871640551114400141, −15.63739373880358176658779270465, −15.09483625535602010655450632543, −13.0605728973842990653730897921, −12.15127649051442137001616639136, −10.89709987910668328017892638349, −10.248837677501446595677957808322, −9.22190805039409663574768876914, −7.8568457587687067507165788813, −7.03358585400019844987330151650, −5.77671345626731398361794857706, −4.7660796705630358287048436319, −3.22821791140358369508240696084, −0.90642663841616088060100676316, 0.447469487872252784030644195783, 2.37056904974004844872293605022, 3.53615176630219137439514228451, 5.281317907760856175862598444642, 6.87761576579703427203987872354, 7.24087488078893897412252585189, 8.64742960642288739722122774265, 9.85208925175774081570391207091, 10.94657104171905846473400357253, 11.67394692680232291389903885344, 12.429286199979757183159026340970, 13.54065161582207012674213070325, 15.53503862455372388172626890575, 16.11631846841551706288324244348, 16.90971002915984485379346425554, 18.406943657645579327026238232551, 18.64337132821892692307273668682, 19.452114336659843086456736956224, 20.65593332746581784025428802674, 21.96737771118408487371563539137, 22.54383509421334949625917204625, 23.713244571329033223552680399097, 24.57740812603640657713657024938, 25.79310349478537364577272884454, 26.69134706212721550413854030568

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