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Properties

Label 1-211-211.114-r0-0-0
Degree $1$
Conductor $211$
Sign $0.114 - 0.993i$
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  + (0.936 − 0.351i)2-s + (−0.963 − 0.266i)3-s + (0.753 − 0.657i)4-s + (−0.963 + 0.266i)5-s + (−0.995 + 0.0896i)6-s + (0.936 + 0.351i)7-s + (0.473 − 0.880i)8-s + (0.858 + 0.512i)9-s + (−0.809 + 0.587i)10-s + (−0.0448 − 0.998i)11-s + (−0.900 + 0.433i)12-s + (0.134 − 0.990i)13-s + 14-s + 15-s + (0.134 − 0.990i)16-s + (−0.691 − 0.722i)17-s + ⋯
L(s)  = 1  + (0.936 − 0.351i)2-s + (−0.963 − 0.266i)3-s + (0.753 − 0.657i)4-s + (−0.963 + 0.266i)5-s + (−0.995 + 0.0896i)6-s + (0.936 + 0.351i)7-s + (0.473 − 0.880i)8-s + (0.858 + 0.512i)9-s + (−0.809 + 0.587i)10-s + (−0.0448 − 0.998i)11-s + (−0.900 + 0.433i)12-s + (0.134 − 0.990i)13-s + 14-s + 15-s + (0.134 − 0.990i)16-s + (−0.691 − 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.114 - 0.993i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.114 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047476364 - 0.9332137209i\)
\(L(\frac12)\) \(\approx\) \(1.047476364 - 0.9332137209i\)
\(L(1)\) \(\approx\) \(1.173315985 - 0.5227551231i\)
\(L(1)\) \(\approx\) \(1.173315985 - 0.5227551231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.936 - 0.351i)T \)
3 \( 1 + (-0.963 - 0.266i)T \)
5 \( 1 + (-0.963 + 0.266i)T \)
7 \( 1 + (0.936 + 0.351i)T \)
11 \( 1 + (-0.0448 - 0.998i)T \)
13 \( 1 + (0.134 - 0.990i)T \)
17 \( 1 + (-0.691 - 0.722i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.0448 + 0.998i)T \)
41 \( 1 + (0.983 + 0.178i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.858 - 0.512i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (0.983 + 0.178i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.473 + 0.880i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.550 + 0.834i)T \)
97 \( 1 + (-0.393 - 0.919i)T \)
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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

−26.90008235480712604403287639682, −26.01993474769946400985321990282, −24.51740493072024073261257655731, −23.652259034877061130273388674770, −23.53885644368698290295656793087, −22.31806647520753351572503211302, −21.48956132129782621606955268950, −20.56127072795737330873460047629, −19.635513284626109992247882983330, −17.952606858495485004007774524161, −17.17912839963782058003093983495, −16.22591422191770812219608465204, −15.38044267242272489331096949874, −14.63642814189181788625067819053, −13.196433099857288219599630901118, −12.24146255894142982303712085287, −11.39089252330964468267059672954, −10.84063893284418655487896403795, −8.97170088265282014654539606550, −7.45057214916615017059201935026, −6.91917170045639845489033311126, −5.355201472263601770382503539156, −4.51331321960370533676411491211, −3.88460918177807474052202083991, −1.7341967914566131689141294142, 0.96995890321230227944131052647, 2.6829773911032813846030059552, 4.07734336158487545152474757866, 5.11495387368520663927489640824, 6.01303885605469165669921818845, 7.23943649092287164711492071945, 8.286206823753434218194825394793, 10.38168482701060482649602729315, 11.16644085420085411650437826801, 11.747407540971641549587370714921, 12.64315032807371139065412071596, 13.790128839327458071277977271273, 14.94691522811392336554134328692, 15.78310881924195310836550277841, 16.66316076259384848675701051619, 18.22054618372173322419586560433, 18.774768015244705828826074334075, 20.014738328083931576435730262278, 20.98680995975904993940354910131, 22.0334456834461407667497590037, 22.75114068277686827682202104153, 23.46951722833616625850903295773, 24.45609476661785541013484304361, 24.82994529381355184444601217207, 26.93389682908002917004442448774

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