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Properties

Label 2-2736-228.11-c1-0-16
Degree $2$
Conductor $2736$
Sign $0.238 - 0.971i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.03 + 1.17i)5-s + 4.52i·7-s + 2.20·11-s + (−1.93 − 3.35i)13-s + (5.29 + 3.05i)17-s + (2.75 + 3.37i)19-s + (−0.557 − 0.966i)23-s + (0.270 + 0.468i)25-s + (3.51 − 2.02i)29-s − 6.92i·31-s + (−5.33 + 9.23i)35-s + 10.9·37-s + (−8.80 − 5.08i)41-s + (9.76 + 5.63i)43-s + (6.43 + 11.1i)47-s + ⋯
L(s)  = 1  + (0.911 + 0.526i)5-s + 1.71i·7-s + 0.666·11-s + (−0.536 − 0.929i)13-s + (1.28 + 0.740i)17-s + (0.632 + 0.774i)19-s + (−0.116 − 0.201i)23-s + (0.0541 + 0.0937i)25-s + (0.652 − 0.376i)29-s − 1.24i·31-s + (−0.900 + 1.56i)35-s + 1.80·37-s + (−1.37 − 0.794i)41-s + (1.48 + 0.859i)43-s + (0.938 + 1.62i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.238 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.357474078\)
\(L(\frac12)\) \(\approx\) \(2.357474078\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-2.75 - 3.37i)T \)
good5 \( 1 + (-2.03 - 1.17i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.52iT - 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 + (1.93 + 3.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.29 - 3.05i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.557 + 0.966i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 + 2.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (8.80 + 5.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.76 - 5.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.43 - 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.32 - 4.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.66 - 6.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.719 + 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.28 - 3.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.87 + 6.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.705 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.92 - 1.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (2.44 - 1.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.76 - 6.52i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.234445771100508306929226585975, −8.083282098289527942248511315771, −7.72772984472798903547881452545, −6.31187028357861897086334098653, −5.89085533075237052003679368779, −5.50655511824838352950810074544, −4.25186309779363273382162115682, −2.96706726243034859320058426960, −2.51041654357022688539094488532, −1.36177365411992899800848010439, 0.852841249915738396147576678410, 1.57442108637266004013747059121, 2.98130455443870506403094807323, 3.96030305176814146860923367248, 4.77228292246451321145814740806, 5.42255356001822683365013057127, 6.58299838224299294043136026060, 7.08254766432743562823098063432, 7.74641618052332630227052593587, 8.830524098770022897184464195462

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