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Properties

Label 2-2736-228.83-c1-0-8
Degree $2$
Conductor $2736$
Sign $-0.331 - 0.943i$
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  + (1.34 − 0.776i)5-s + 2.95i·7-s − 3.57·11-s + (2.39 − 4.14i)13-s + (−3.37 + 1.95i)17-s + (−2.61 − 3.48i)19-s + (−2.64 + 4.57i)23-s + (−1.29 + 2.24i)25-s + (7.08 + 4.09i)29-s + 7.52i·31-s + (2.29 + 3.97i)35-s + 2.32·37-s + (−3.70 + 2.14i)41-s + (3.48 − 2.01i)43-s + (−4.08 + 7.06i)47-s + ⋯
L(s)  = 1  + (0.601 − 0.347i)5-s + 1.11i·7-s − 1.07·11-s + (0.663 − 1.14i)13-s + (−0.819 + 0.473i)17-s + (−0.599 − 0.800i)19-s + (−0.550 + 0.954i)23-s + (−0.259 + 0.448i)25-s + (1.31 + 0.759i)29-s + 1.35i·31-s + (0.387 + 0.671i)35-s + 0.381·37-s + (−0.579 + 0.334i)41-s + (0.531 − 0.306i)43-s + (−0.595 + 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.184038713\)
\(L(\frac12)\) \(\approx\) \(1.184038713\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 3.48i)T \)
good5 \( 1 + (-1.34 + 0.776i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.95iT - 7T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 + (-2.39 + 4.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.37 - 1.95i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.64 - 4.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.08 - 4.09i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.52iT - 31T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + (3.70 - 2.14i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.48 + 2.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.08 - 7.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.03 - 1.17i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.14 - 5.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.02 - 5.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.72 + 4.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.18 + 8.98i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.60 + 4.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.9T + 83T^{2} \)
89 \( 1 + (7.70 + 4.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.08 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.801048426514025911614509557076, −8.561424574500219077774020561615, −7.68212831922480098387524994343, −6.63148138298472245240012226685, −5.77310062984244384247435806464, −5.38538026539869749174192614143, −4.52050652433076104526594343369, −3.15456852189224522332532986738, −2.49754954327410948516062494445, −1.38827435694262315912341967623, 0.36539031229742724082296518166, 1.91278997141321175155091978023, 2.64281854378359230169453104338, 4.05420145116938670147412145770, 4.39323648322228325209820918329, 5.60348022993924861744113558434, 6.52607792734871298543742649104, 6.80600950770646157858602775322, 8.014612590535962963932606887212, 8.363288546017747771671287542399

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