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Properties

Label 1-211-211.82-r0-0-0
Degree $1$
Conductor $211$
Sign $0.684 + 0.728i$
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  + (0.858 + 0.512i)2-s + (−0.393 − 0.919i)3-s + (0.473 + 0.880i)4-s + (−0.393 + 0.919i)5-s + (0.134 − 0.990i)6-s + (0.858 − 0.512i)7-s + (−0.0448 + 0.998i)8-s + (−0.691 + 0.722i)9-s + (−0.809 + 0.587i)10-s + (0.753 − 0.657i)11-s + (0.623 − 0.781i)12-s + (−0.550 + 0.834i)13-s + 14-s + 15-s + (−0.550 + 0.834i)16-s + (0.936 + 0.351i)17-s + ⋯
L(s)  = 1  + (0.858 + 0.512i)2-s + (−0.393 − 0.919i)3-s + (0.473 + 0.880i)4-s + (−0.393 + 0.919i)5-s + (0.134 − 0.990i)6-s + (0.858 − 0.512i)7-s + (−0.0448 + 0.998i)8-s + (−0.691 + 0.722i)9-s + (−0.809 + 0.587i)10-s + (0.753 − 0.657i)11-s + (0.623 − 0.781i)12-s + (−0.550 + 0.834i)13-s + 14-s + 15-s + (−0.550 + 0.834i)16-s + (0.936 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.572223143 + 0.6800293129i\)
\(L(\frac12)\) \(\approx\) \(1.572223143 + 0.6800293129i\)
\(L(1)\) \(\approx\) \(1.459932723 + 0.3715353860i\)
\(L(1)\) \(\approx\) \(1.459932723 + 0.3715353860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (0.858 + 0.512i)T \)
3 \( 1 + (-0.393 - 0.919i)T \)
5 \( 1 + (-0.393 + 0.919i)T \)
7 \( 1 + (0.858 - 0.512i)T \)
11 \( 1 + (0.753 - 0.657i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (0.936 + 0.351i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (0.753 + 0.657i)T \)
41 \( 1 + (-0.963 + 0.266i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.691 - 0.722i)T \)
53 \( 1 + (0.473 - 0.880i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.0448 - 0.998i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.995 + 0.0896i)T \)
97 \( 1 + (0.983 - 0.178i)T \)
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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

−27.053553635890097764693602767619, −25.2584662260319906098493684967, −24.57285645313346658942010154666, −23.49711607574758371431091575056, −22.7482679448235457847492156674, −21.75566605127409212162380275356, −21.02205270999706394201554799561, −20.256560830739088021624674840079, −19.49060449776072081952238351516, −17.81894730555174865601596152801, −16.91484119721912712913139026476, −15.62954272772313995524054541194, −15.14379716683766874279920490527, −14.1383594521851267613054993650, −12.66871765884789513194284593224, −11.861499804471314965519599321721, −11.26909821989098285895399831425, −9.88117919583357858216613478116, −9.10861945607653871105113138544, −7.54901904371055182393704544220, −5.7608913511465431605695559551, −4.99659189785895252391550433672, −4.308873031292565558807388916686, −2.97918042549253212231088157870, −1.22171054872651388135143513509, 1.732180065001980564568820930309, 3.19763360939598405122728485782, 4.421066469920091850487159563446, 5.796507883019477265686672048781, 6.73391338666329167073267752537, 7.52938701864582841479529662005, 8.36773074980155695919319971075, 10.56838402494797193718274448162, 11.63214698019322300418089329623, 12.02999549993471815440235132956, 13.49093433410153675898429128336, 14.39492123213288864969743456946, 14.72680574155312298746375953790, 16.592816908993516631593087579081, 16.94107789370493663458374679055, 18.2645781773780787892600083252, 19.075027903104315470678281883294, 20.254941345511130155541011564594, 21.50602777444543802814284537132, 22.39952221584918861226616781387, 23.167218330183320165228982706324, 23.998889945574270711764581868079, 24.56103947747966422427500384020, 25.661232164228568092872659680319, 26.678935384761191271532704496412

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