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Properties

Label 1-2808-2808.1181-r0-0-0
Degree $1$
Conductor $2808$
Sign $0.856 - 0.516i$
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  + (−0.342 − 0.939i)5-s + (−0.642 − 0.766i)7-s + (−0.642 − 0.766i)11-s + 17-s + (0.866 + 0.5i)19-s + (0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)29-s + (0.984 + 0.173i)31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + (0.342 + 0.939i)41-s + (0.173 + 0.984i)43-s + (−0.984 + 0.173i)47-s + (−0.173 + 0.984i)49-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)5-s + (−0.642 − 0.766i)7-s + (−0.642 − 0.766i)11-s + 17-s + (0.866 + 0.5i)19-s + (0.766 + 0.642i)23-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)29-s + (0.984 + 0.173i)31-s + (−0.5 + 0.866i)35-s + (0.866 − 0.5i)37-s + (0.342 + 0.939i)41-s + (0.173 + 0.984i)43-s + (−0.984 + 0.173i)47-s + (−0.173 + 0.984i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(13.0402\)
Root analytic conductor: \(13.0402\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2808,\ (0:\ ),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.379316488 - 0.3840863143i\)
\(L(\frac12)\) \(\approx\) \(1.379316488 - 0.3840863143i\)
\(L(1)\) \(\approx\) \(0.9688453631 - 0.2119159884i\)
\(L(1)\) \(\approx\) \(0.9688453631 - 0.2119159884i\)

Euler product

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

−19.13061055189184494748888765520, −18.58781349408708213446371682604, −18.083357664942648026529918317760, −17.228022623991887028254792755552, −16.33089306539136281463029397169, −15.50780273589482443842427384817, −15.24350591708707793194424574421, −14.44322702393146826807034272366, −13.599718420865413430222340196428, −12.85745009903808532773863737554, −12.00816120526806129641877888073, −11.61949904489784804109265980358, −10.52825535851481210042638729629, −10.007058381809505687666123232282, −9.34542833470572931184473559681, −8.32595284015975541767749887842, −7.57250416765481454348295941973, −6.92911797056603394491736603305, −6.145464582282768280025794237496, −5.372594813416441501694130844, −4.48468897819658102487961434624, −3.42382937453393800607317536973, −2.772035514315288304048758219963, −2.19355915687757510215044607232, −0.65803542162448144914759848302, 0.82064242003241453692168180600, 1.29580788219394926150149755368, 2.92877551411081813664069317589, 3.41037155581669654603897955752, 4.33752494850113097016373719441, 5.2000885053721537134137914051, 5.803341494746869314218164006228, 6.80333070010863562920546459381, 7.773820339154307959293301265658, 8.066130297519821061036011519479, 9.16649375469545893504737984517, 9.733551312833685015370089393, 10.513447220682356546042968882120, 11.353402611411968184240861188151, 12.06689881633048651545678682429, 12.91905730156350451969494719148, 13.31087458542186535457138599581, 14.08904914947173670685369904481, 14.92199753374100000091011770494, 16.02823299703970090129127489681, 16.26172449652724229030067222063, 16.78721403737953753113516005249, 17.70636350605582352452047330525, 18.47957049305493125693625443677, 19.44599548865823675460298239203

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