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Properties

Label 2-2736-228.83-c1-0-34
Degree $2$
Conductor $2736$
Sign $0.511 + 0.859i$
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  + (1.62 − 0.935i)5-s − 2.07i·7-s + 5.79·11-s + (3.25 − 5.63i)13-s + (3.56 − 2.06i)17-s + (2.66 + 3.44i)19-s + (2.02 − 3.50i)23-s + (−0.748 + 1.29i)25-s + (−2.90 − 1.67i)29-s + 5.15i·31-s + (−1.94 − 3.36i)35-s − 6.99·37-s + (−9.12 + 5.27i)41-s + (−0.638 + 0.368i)43-s + (−4.95 + 8.58i)47-s + ⋯
L(s)  = 1  + (0.724 − 0.418i)5-s − 0.785i·7-s + 1.74·11-s + (0.901 − 1.56i)13-s + (0.865 − 0.499i)17-s + (0.611 + 0.791i)19-s + (0.422 − 0.731i)23-s + (−0.149 + 0.259i)25-s + (−0.538 − 0.310i)29-s + 0.925i·31-s + (−0.328 − 0.569i)35-s − 1.15·37-s + (−1.42 + 0.823i)41-s + (−0.0973 + 0.0561i)43-s + (−0.723 + 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.511 + 0.859i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.511 + 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.572054764\)
\(L(\frac12)\) \(\approx\) \(2.572054764\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.66 - 3.44i)T \)
good5 \( 1 + (-1.62 + 0.935i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.07iT - 7T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 + (-3.25 + 5.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.56 + 2.06i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.02 + 3.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.90 + 1.67i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.15iT - 31T^{2} \)
37 \( 1 + 6.99T + 37T^{2} \)
41 \( 1 + (9.12 - 5.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.638 - 0.368i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.95 - 8.58i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.83 + 2.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.37 - 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.20 + 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 5.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.81 + 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.73 - 8.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.40 - 5.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.44T + 83T^{2} \)
89 \( 1 + (1.89 + 1.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.380 + 0.658i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.656802010733101906868908923729, −8.060755351073385211984031477520, −7.10999441268890092246166263609, −6.38531033356808250921095929039, −5.60981128841426663115704357133, −4.91472985966516429375958138648, −3.67828101035492570474069242939, −3.27740314713431790351123617307, −1.53539969886895410882982441196, −0.980498855749355197580391599547, 1.42375341367464775558816710631, 2.05696240501018189797195453571, 3.41237733513803829849160094916, 4.00520588760838559699975194023, 5.24824272948291473456535650083, 5.95682455693593461143569300201, 6.66279147904977188870473837119, 7.13119925742604166245683453315, 8.499661806615301330456535690039, 9.015911224527458786517075158579

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