Properties

Label 1-2736-2736.11-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.937 - 0.347i$
Analytic cond. $12.7059$
Root an. cond. $12.7059$
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  i·5-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s i·29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)35-s + i·37-s + 41-s + (0.866 + 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  i·5-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)23-s − 25-s i·29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)35-s + i·37-s + 41-s + (0.866 + 0.5i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.937 - 0.347i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.569615517 - 0.2810668528i\)
\(L(\frac12)\) \(\approx\) \(1.569615517 - 0.2810668528i\)
\(L(1)\) \(\approx\) \(1.058417093 - 0.1916030512i\)
\(L(1)\) \(\approx\) \(1.058417093 - 0.1916030512i\)

Euler product

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

−19.29977862629849120976417874144, −18.65213563143784226287934842780, −17.911508389829981085426743348744, −17.35065688873470575714689134624, −16.444141912333014942029736649026, −15.61715411774772812298437168075, −15.05220135555489471380471746054, −14.406413606261803964353703365422, −13.855478932838733046285855813875, −12.64049637002860924461937503560, −12.23226631506830716466122543866, −11.51631235265600428843924631141, −10.653268574446956005946344664050, −9.83837905734989561462240657897, −9.3946077843222108034693511694, −8.483355190094750975712868158118, −7.41088807867329742813834984995, −6.969521312754485274929156313229, −6.12095138938908318042526021254, −5.43044473469530829928936345414, −4.43274847572166026906081023284, −3.4889954904949966014357861962, −2.615941740785214932506427127685, −2.231964746178436285978482009352, −0.669924341940358828789316189620, 0.873848602542078945935534119794, 1.427595334407275063073741407529, 2.69135772922044062861348801274, 3.82154700898538097951217662983, 4.187709854469562710403790569178, 5.18574570486374617948417789246, 5.98356277681535513803878677293, 6.81465098175711472276979567809, 7.592704474444893349428974568074, 8.351513758634255288810169620402, 9.31428196939237381106082117557, 9.60846047682564536346536491417, 10.59594131887673005146874261342, 11.41539967111055286689963732407, 12.21685074670112274367598234427, 12.76026619473650519191470406958, 13.56799091957209188528072937518, 14.13311882105736124402731954419, 14.952072826469596714290560870054, 15.86139075168477493840612614659, 16.608119361021812265273750824686, 17.07501711751345743819883178859, 17.38679882910622797239032531079, 18.7358360514629437509455343001, 19.43331706233432358770143306147

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