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Properties

Label 1-211-211.140-r1-0-0
Degree $1$
Conductor $211$
Sign $-0.998 - 0.0555i$
Analytic cond. $22.6750$
Root an. cond. $22.6750$
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  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.587i)11-s − 12-s + (0.309 + 0.951i)13-s + 14-s − 15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.809 − 0.587i)11-s − 12-s + (0.309 + 0.951i)13-s + 14-s − 15-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $-0.998 - 0.0555i$
Analytic conductor: \(22.6750\)
Root analytic conductor: \(22.6750\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (1:\ ),\ -0.998 - 0.0555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008022225249 - 0.2888322241i\)
\(L(\frac12)\) \(\approx\) \(0.008022225249 - 0.2888322241i\)
\(L(1)\) \(\approx\) \(0.7357014378 - 0.05636468648i\)
\(L(1)\) \(\approx\) \(0.7357014378 - 0.05636468648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−26.97572674545010975561178390072, −26.037597710994284444925871944487, −25.648741966852110144984068594923, −24.05850814117833232221399870292, −22.63035984653252857657591984270, −22.16234858620611446783055319518, −21.133207485323914579438887986565, −20.160245673711754048081639126417, −19.53539040610739567203447961058, −18.58949335993028711813360818652, −17.85839560483038647978290809153, −16.12357232581916258094084790782, −15.358596888378076520188273441744, −14.50469312878794481391224547135, −13.05655003336141594857640347803, −12.39099615865749424345901467759, −10.90167430225923980769365308635, −10.42445435252455139228818629039, −9.12181234561071007792994107020, −8.35243597022552019738305580038, −7.36988300521764143949240276712, −5.277518228072710328923793879863, −4.01053519796516683078694279127, −3.01626804477802331786663797632, −2.23537658968310556266674839348, 0.1018567434268933484985691265, 1.34039900408943480192912916684, 3.484190345501522194667375322609, 4.45634933028339734764053755134, 6.01932846391223679223876531128, 7.37680399396293269090276608971, 7.73861205197603473486448538193, 8.87427712840219816972689619439, 9.7395502657985656917863922650, 11.29106089522413022778627006311, 12.76410038735106769056667818384, 13.646765140670360474895019676892, 14.267070015744911618999416758314, 15.71397572516316817237134297888, 16.16489072257013859570091790754, 17.293911726809437230888521452346, 18.577420782314102660314465751, 19.138372511901219021013069595133, 20.11516896660463895554800134015, 20.9807592617333081564900967961, 22.7579601184967503405084824423, 23.66861009972324032388503139431, 24.037028034433728711646255986347, 25.0210158496454793419375330335, 26.08289918388097591136302350713

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