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Properties

Label 2-2736-228.83-c1-0-32
Degree $2$
Conductor $2736$
Sign $-0.336 + 0.941i$
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  + (−1.02 + 0.592i)5-s − 4.00i·7-s + 3.72·11-s + (−0.163 + 0.284i)13-s + (2.92 − 1.69i)17-s + (−3.68 − 2.32i)19-s + (1.20 − 2.08i)23-s + (−1.79 + 3.11i)25-s + (−0.884 − 0.510i)29-s − 0.252i·31-s + (2.37 + 4.11i)35-s − 2.26·37-s + (6.04 − 3.48i)41-s + (−0.904 + 0.522i)43-s + (3.48 − 6.03i)47-s + ⋯
L(s)  = 1  + (−0.459 + 0.265i)5-s − 1.51i·7-s + 1.12·11-s + (−0.0454 + 0.0787i)13-s + (0.710 − 0.410i)17-s + (−0.846 − 0.533i)19-s + (0.251 − 0.435i)23-s + (−0.359 + 0.622i)25-s + (−0.164 − 0.0948i)29-s − 0.0452i·31-s + (0.401 + 0.695i)35-s − 0.372·37-s + (0.943 − 0.544i)41-s + (−0.137 + 0.0796i)43-s + (0.508 − 0.880i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.336 + 0.941i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.325507159\)
\(L(\frac12)\) \(\approx\) \(1.325507159\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.68 + 2.32i)T \)
good5 \( 1 + (1.02 - 0.592i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.00iT - 7T^{2} \)
11 \( 1 - 3.72T + 11T^{2} \)
13 \( 1 + (0.163 - 0.284i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.92 + 1.69i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.20 + 2.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.884 + 0.510i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.252iT - 31T^{2} \)
37 \( 1 + 2.26T + 37T^{2} \)
41 \( 1 + (-6.04 + 3.48i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.904 - 0.522i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.48 + 6.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.8 + 6.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.581 - 1.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.11 - 7.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.16 - 4.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.12 - 1.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.05 + 1.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.56 + 3.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.14T + 83T^{2} \)
89 \( 1 + (3.86 + 2.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.12 + 7.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.550700863161715343766000932307, −7.68758751421257918728269493757, −7.04282920549487144450644241462, −6.61126244049405881782550803108, −5.48061418804460142448188345396, −4.33168100527933488375001051385, −3.94972883006662998237401995189, −3.03298550761123978030563112757, −1.57989804577823681585270345209, −0.45594450782165926731438478514, 1.35726995009335127002902461813, 2.39854928650580591802188778271, 3.47001549706225025629842298886, 4.27836505748713358148999652555, 5.24339650480272732213046683289, 6.05182683892740260007931996227, 6.54723895664196470302164841469, 7.84151086865110400602151136249, 8.224833119492484351044123706251, 9.189300753025969900221775949116

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