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Properties

Label 2-2736-19.7-c1-0-33
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  + (−1 − 1.73i)5-s − 3·7-s + 6·11-s + (0.5 − 0.866i)13-s + (−1 − 1.73i)17-s + (4 + 1.73i)19-s + (0.500 − 0.866i)25-s + (−1 + 1.73i)29-s + 31-s + (3 + 5.19i)35-s − 7·37-s + (−0.5 − 0.866i)43-s + 2·49-s + (−2 + 3.46i)53-s + (−6 − 10.3i)55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s − 1.13·7-s + 1.80·11-s + (0.138 − 0.240i)13-s + (−0.242 − 0.420i)17-s + (0.917 + 0.397i)19-s + (0.100 − 0.173i)25-s + (−0.185 + 0.321i)29-s + 0.179·31-s + (0.507 + 0.878i)35-s − 1.15·37-s + (−0.0762 − 0.132i)43-s + 0.285·49-s + (−0.274 + 0.475i)53-s + (−0.809 − 1.40i)55-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: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.325229045\)
\(L(\frac12)\) \(\approx\) \(1.325229045\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.5 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.815751935529598399336026629237, −7.929373110075289372246179410398, −6.94953495799220625386005164377, −6.46676353902859107088307218351, −5.55342501522582746160853348767, −4.59947917396128513003517154896, −3.74982591028905006728197135366, −3.14966209925132612810778793289, −1.59536403484124578836069784750, −0.50175226794472828149829641733, 1.15352962241163900241119319056, 2.55189422188455684418291350211, 3.68330849469412559022062920112, 3.79026423806617108421397104815, 5.18886732773353441794442939639, 6.28225212430359841448048758840, 6.74235916654225749158340644987, 7.21891069205695543263464810000, 8.343902814544177251246791840366, 9.166364681294549300852532771614

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