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Properties

Label 2-2736-19.11-c1-0-10
Degree $2$
Conductor $2736$
Sign $-0.0977 - 0.995i$
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  + (−0.5 + 0.866i)5-s − 4·11-s + (0.5 + 0.866i)13-s + (1.5 − 2.59i)17-s + (4 − 1.73i)19-s + (−2.5 − 4.33i)23-s + (2 + 3.46i)25-s + (3.5 + 6.06i)29-s − 4·31-s + 10·37-s + (−2.5 + 4.33i)41-s + (−2.5 + 4.33i)43-s + (3.5 + 6.06i)47-s − 7·49-s + (5.5 + 9.52i)53-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s − 1.20·11-s + (0.138 + 0.240i)13-s + (0.363 − 0.630i)17-s + (0.917 − 0.397i)19-s + (−0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s + (0.649 + 1.12i)29-s − 0.718·31-s + 1.64·37-s + (−0.390 + 0.676i)41-s + (−0.381 + 0.660i)43-s + (0.510 + 0.884i)47-s − 49-s + (0.755 + 1.30i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.165823446\)
\(L(\frac12)\) \(\approx\) \(1.165823446\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.5 + 4.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 - 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.5 - 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.054705403434683266982623551208, −8.140789053064287855912961104240, −7.52076972315539384232848272679, −6.88361181659730053854982346490, −5.92280105664678155302966185313, −5.11424772070726881081185604944, −4.39502527415872780641166664417, −3.12014877904489795241903560323, −2.67666825197449309790647824329, −1.15918922571281169518201343187, 0.41216880717598192294647371678, 1.79115706186670023341258478109, 2.91629299416387880212151279329, 3.80394392294044327547846539700, 4.75054825821794167386846614079, 5.55715079039399319366086991756, 6.13773872994092676326782427873, 7.36794017330594851920613005949, 7.87920015256553347887430123904, 8.466716316120410172825665480837

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