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Properties

Label 1-2808-2808.133-r0-0-0
Degree $1$
Conductor $2808$
Sign $0.308 + 0.951i$
Analytic cond. $13.0402$
Root an. cond. $13.0402$
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  + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)7-s + (0.939 + 0.342i)11-s + 17-s + (0.5 − 0.866i)19-s + (−0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.766 + 0.642i)29-s + (0.766 + 0.642i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.766 − 0.642i)47-s + (0.766 + 0.642i)49-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)5-s + (−0.939 − 0.342i)7-s + (0.939 + 0.342i)11-s + 17-s + (0.5 − 0.866i)19-s + (−0.939 + 0.342i)23-s + (−0.939 − 0.342i)25-s + (−0.766 + 0.642i)29-s + (0.766 + 0.642i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (0.766 − 0.642i)47-s + (0.766 + 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(13.0402\)
Root analytic conductor: \(13.0402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (0:\ ),\ 0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126526611 + 0.8189808560i\)
\(L(\frac12)\) \(\approx\) \(1.126526611 + 0.8189808560i\)
\(L(1)\) \(\approx\) \(0.9757427192 + 0.2156372302i\)
\(L(1)\) \(\approx\) \(0.9757427192 + 0.2156372302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 - T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−18.95677834661454382133102297353, −18.67416679883559660076482207842, −17.45864833470178186549448965596, −16.8306481225446736416774519935, −16.26650610636486814883132021791, −15.81550072344999369072969303661, −14.79909926437807874588021803756, −14.10701978718747681396162948528, −13.29650530089624369796572782065, −12.61559863524373267105367303476, −11.96799056608683379627197707218, −11.54308207475564553783300342918, −10.22877021282894954750062847801, −9.599275690775510779418690818715, −9.13065764581056792034324501633, −8.15565194711618899746964251809, −7.66243312884805011129824026843, −6.41227812545320123259114974436, −5.92189742882847748717780044532, −5.182570292190177920100827391357, −3.9746138920012481418512213159, −3.68625982436427910417735736516, −2.50729261401640662535490128485, −1.45496510055100970334085565394, −0.55347404725185641437389369596, 0.910964697427867870089616186117, 2.04828053029837333867942736455, 3.122271673398253882667583778285, 3.52715797503841309128768000816, 4.39354668108583883343790038441, 5.54897005764866786770091939468, 6.34810331692837213446061933114, 6.98078360598933951104053457744, 7.4870176039106925186621405103, 8.514737767735391442705947253730, 9.56865771658664596390288089892, 9.91441349419325676872105133094, 10.69453386984715588696096766179, 11.61827802254556008397549156301, 12.07164059050181850904989323369, 13.02134055989535029044097579044, 13.82128296223130222267574284419, 14.36228606654720968463675332976, 15.094395873167739371707987338426, 15.8456045643728965758157420261, 16.46863633000903908112672549041, 17.31484012585025320207412479003, 17.922194761534261310976034227640, 18.804805113746341113288034549062, 19.28219675342771035974671210459

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