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Properties

Label 1-211-211.25-r0-0-0
Degree $1$
Conductor $211$
Sign $0.591 + 0.806i$
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  + (−0.0448 + 0.998i)2-s + (0.936 + 0.351i)3-s + (−0.995 − 0.0896i)4-s + (0.936 − 0.351i)5-s + (−0.393 + 0.919i)6-s + (−0.0448 − 0.998i)7-s + (0.134 − 0.990i)8-s + (0.753 + 0.657i)9-s + (0.309 + 0.951i)10-s + (−0.550 − 0.834i)11-s + (−0.900 − 0.433i)12-s + (0.983 + 0.178i)13-s + 14-s + 15-s + (0.983 + 0.178i)16-s + (0.473 + 0.880i)17-s + ⋯
L(s)  = 1  + (−0.0448 + 0.998i)2-s + (0.936 + 0.351i)3-s + (−0.995 − 0.0896i)4-s + (0.936 − 0.351i)5-s + (−0.393 + 0.919i)6-s + (−0.0448 − 0.998i)7-s + (0.134 − 0.990i)8-s + (0.753 + 0.657i)9-s + (0.309 + 0.951i)10-s + (−0.550 − 0.834i)11-s + (−0.900 − 0.433i)12-s + (0.983 + 0.178i)13-s + 14-s + 15-s + (0.983 + 0.178i)16-s + (0.473 + 0.880i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.591 + 0.806i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.591 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455802171 + 0.7380477286i\)
\(L(\frac12)\) \(\approx\) \(1.455802171 + 0.7380477286i\)
\(L(1)\) \(\approx\) \(1.304071917 + 0.5533642298i\)
\(L(1)\) \(\approx\) \(1.304071917 + 0.5533642298i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.0448 + 0.998i)T \)
3 \( 1 + (0.936 + 0.351i)T \)
5 \( 1 + (0.936 - 0.351i)T \)
7 \( 1 + (-0.0448 - 0.998i)T \)
11 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (0.473 + 0.880i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.983 + 0.178i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (-0.550 + 0.834i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.753 - 0.657i)T \)
53 \( 1 + (-0.995 + 0.0896i)T \)
59 \( 1 + (-0.691 + 0.722i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (0.134 + 0.990i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.963 + 0.266i)T \)
97 \( 1 + (0.858 - 0.512i)T \)
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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

−26.45345660429030130935922928565, −25.52397411010688379503820366749, −25.15263843857991478403998330621, −23.58025363516803694410586087710, −22.58098643468249156793189858192, −21.461395868783509353453247185759, −20.907430743790763120999214379828, −20.10961537376559250921311472251, −18.698149319413353201195089429295, −18.45610539970442531400137362940, −17.592594723850937510242879663175, −15.7922544907217853851930920505, −14.57124362737387371687486073611, −13.905443789808847349716013091061, −12.82586083296832505186629177696, −12.23582694328666471236057975706, −10.65350769627730114575191883764, −9.76476366695530287519676478324, −8.92958150772939757736803505158, −7.97568039355715605757470225173, −6.35577644346500732434856186262, −5.058199540719058911924650563601, −3.47259913640096907200969887209, −2.427383624300296608435985168830, −1.732064424286780001525779623404, 1.450992907404523546246106337806, 3.38644386241473167321964763286, 4.42347912032920929298365662807, 5.692027694633160381209964168558, 6.78748099310402165018081037296, 8.1522591738442957947692238243, 8.70122741530201390275172426291, 9.931530061703092755665824733465, 10.6085525602728939895052320369, 13.00251896366110307964128269152, 13.56586260940551018890993715105, 14.15930684586455505481096185118, 15.33604126748668201480977787798, 16.33544969817030034371926318161, 17.00175620364346869623214343726, 18.16472982208070531286274742035, 19.14740650690246586499462037662, 20.28592440680849892820945318661, 21.36019087977697339622022648249, 21.90371751511764546370734180680, 23.59626788513803906614390506514, 23.96516404775656286100934483082, 25.28634556268311355055550441165, 25.84229046927616417783859686913, 26.4229151893250779500878214455

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