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Properties

Label 1-208-208.163-r0-0-0
Degree $1$
Conductor $208$
Sign $0.913 + 0.406i$
Analytic cond. $0.965947$
Root an. cond. $0.965947$
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  + (−0.866 + 0.5i)3-s + 5-s + (0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s + i·27-s + (0.866 − 0.5i)29-s − i·31-s + (0.866 + 0.5i)33-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + 5-s + (0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 21-s + (0.5 + 0.866i)23-s + 25-s + i·27-s + (0.866 − 0.5i)29-s − i·31-s + (0.866 + 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(0.965947\)
Root analytic conductor: \(0.965947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 208,\ (0:\ ),\ 0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.121874590 + 0.2382332461i\)
\(L(\frac12)\) \(\approx\) \(1.121874590 + 0.2382332461i\)
\(L(1)\) \(\approx\) \(1.031392472 + 0.1472831237i\)
\(L(1)\) \(\approx\) \(1.031392472 + 0.1472831237i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−26.69034650162647826324746024796, −25.58688365761183548271605549385, −24.756642407903903040792387877980, −23.73391406606973223176405074262, −23.15564889377895152062573727945, −21.88016714125440220920745133572, −21.233768513257696934446826689512, −20.146696575065606585473428458486, −18.81063807741233942109165365185, −17.76194376166489873993162110509, −17.44407464353920529084167780078, −16.46875649312290960262361974607, −15.01476505838777988204576233666, −13.96606537179948830244848906931, −12.97525401761473292830702189521, −12.18454776272636455172903951243, −10.66817703031284732110098017012, −10.41004362384369411081368071501, −8.748585174777904578281604075555, −7.43499171747751970234400491979, −6.53880115536218351917951280443, −5.32413777010439055154854746541, −4.54857480724010740576398996885, −2.33990282091787618216307895099, −1.27830579462007425073076978759, 1.3264397751997313192077514541, 2.914309564393433369278902152518, 4.640479270504461283345283927454, 5.545582701330099384576936349044, 6.24208994868982283080315576631, 7.88273689746341401600325899421, 9.18050169253983942564911068371, 10.12478431598719873574676317052, 11.11040814926611434651970579960, 11.9534659075740590579712965475, 13.21730898891879753288634764538, 14.285644014377111719380472650398, 15.32312585208377110007905570443, 16.4383934305011322439809534294, 17.23336408979837195725461083143, 18.1335275673050800411986186561, 18.82589477157399314062731931960, 20.76888403906694671806166518225, 21.1834967013115164996973917297, 21.92916553716692210478048628342, 22.97729883408824640352440789239, 23.992357205652160207731409803385, 24.861675034666661038828194217286, 25.86138333764942529879476200438, 27.130036499633553812277988951476

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