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Properties

Label 1-211-211.183-r0-0-0
Degree $1$
Conductor $211$
Sign $0.932 - 0.362i$
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  + (0.473 + 0.880i)2-s + (−0.691 + 0.722i)3-s + (−0.550 + 0.834i)4-s + (−0.691 − 0.722i)5-s + (−0.963 − 0.266i)6-s + (0.473 − 0.880i)7-s + (−0.995 − 0.0896i)8-s + (−0.0448 − 0.998i)9-s + (0.309 − 0.951i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (−0.393 − 0.919i)13-s + 14-s + 15-s + (−0.393 − 0.919i)16-s + (0.753 + 0.657i)17-s + ⋯
L(s)  = 1  + (0.473 + 0.880i)2-s + (−0.691 + 0.722i)3-s + (−0.550 + 0.834i)4-s + (−0.691 − 0.722i)5-s + (−0.963 − 0.266i)6-s + (0.473 − 0.880i)7-s + (−0.995 − 0.0896i)8-s + (−0.0448 − 0.998i)9-s + (0.309 − 0.951i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (−0.393 − 0.919i)13-s + 14-s + 15-s + (−0.393 − 0.919i)16-s + (0.753 + 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.932 - 0.362i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.932 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7056177605 - 0.1322799454i\)
\(L(\frac12)\) \(\approx\) \(0.7056177605 - 0.1322799454i\)
\(L(1)\) \(\approx\) \(0.7874552466 + 0.2166709970i\)
\(L(1)\) \(\approx\) \(0.7874552466 + 0.2166709970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.473 + 0.880i)T \)
3 \( 1 + (-0.691 + 0.722i)T \)
5 \( 1 + (-0.691 - 0.722i)T \)
7 \( 1 + (0.473 - 0.880i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (-0.393 - 0.919i)T \)
17 \( 1 + (0.753 + 0.657i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (0.134 + 0.990i)T \)
41 \( 1 + (0.858 - 0.512i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.0448 + 0.998i)T \)
53 \( 1 + (-0.550 - 0.834i)T \)
59 \( 1 + (0.858 - 0.512i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (-0.995 + 0.0896i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.983 - 0.178i)T \)
97 \( 1 + (0.936 - 0.351i)T \)
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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

−27.208401939805357153079617586596, −25.718768218918174421574880117865, −24.52547625038533512827724780125, −23.57990070349629269467804907748, −23.047563045532926750501942094577, −22.03433415790921401533772870673, −21.4154372440471327746529436833, −19.88257471126438142612689090384, −19.20524573583066082932615166842, −18.28975128127332184132923229963, −17.76769658433586404133974449731, −16.115352832968908279773267774298, −14.8435269228963626144607190356, −14.21644422809879598406085088919, −12.76135362398241801030057368105, −11.96633289103224612965725644490, −11.48401874968815983722470108033, −10.43326374228552614838928214673, −9.110974925201463841584110726615, −7.59875925993208940259566075357, −6.55753821816887148351409087905, −5.30044146680655949448197714842, −4.30876554653403024057273851213, −2.662151675576842296097224145, −1.70763987610590175475937770724, 0.51426668541345853003230278183, 3.619891457084962100183204209372, 4.224335682560780378324838911531, 5.32809025908941075884941749333, 6.21959166523791741021831590818, 7.78569645330063178667033756709, 8.39606904481031648017700609076, 9.880038697541187785070287197190, 11.08263331707216065538545671161, 12.13285626451072933394527307181, 13.03446489385368302626371355749, 14.37500760519182567125738193321, 15.20717905236280474369121758978, 16.2193378446187729776739801206, 16.89148521534098782226432618111, 17.42539191422434353674261920428, 18.93501891539765120454863398259, 20.458488934411688418365392152082, 21.05437179832487120101493540731, 22.20393869445545071504676092485, 23.008280309390272196567628639462, 23.89842122097590491919048624186, 24.30575454258386758702610405892, 25.737157307344813136192206547640, 26.8574277100950574439511600112

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