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Properties

Label 2-2736-76.31-c1-0-21
Degree $2$
Conductor $2736$
Sign $0.545 - 0.838i$
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  + (1.66 + 2.87i)5-s − 2.71i·7-s + 0.985i·11-s + (4.33 + 2.50i)13-s + (2.51 + 4.35i)17-s + (−0.193 − 4.35i)19-s + (3.68 + 2.12i)23-s + (−3.01 + 5.22i)25-s + (−5.83 − 3.36i)29-s − 2.32·31-s + (7.81 − 4.51i)35-s + 8.27i·37-s + (9.96 − 5.75i)41-s + (−9.48 + 5.47i)43-s + (6.41 + 3.70i)47-s + ⋯
L(s)  = 1  + (0.742 + 1.28i)5-s − 1.02i·7-s + 0.297i·11-s + (1.20 + 0.694i)13-s + (0.609 + 1.05i)17-s + (−0.0443 − 0.999i)19-s + (0.769 + 0.444i)23-s + (−0.602 + 1.04i)25-s + (−1.08 − 0.625i)29-s − 0.416·31-s + (1.32 − 0.762i)35-s + 1.36i·37-s + (1.55 − 0.898i)41-s + (−1.44 + 0.834i)43-s + (0.935 + 0.540i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.259236593\)
\(L(\frac12)\) \(\approx\) \(2.259236593\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.193 + 4.35i)T \)
good5 \( 1 + (-1.66 - 2.87i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.71iT - 7T^{2} \)
11 \( 1 - 0.985iT - 11T^{2} \)
13 \( 1 + (-4.33 - 2.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.51 - 4.35i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.68 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.83 + 3.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 - 8.27iT - 37T^{2} \)
41 \( 1 + (-9.96 + 5.75i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.48 - 5.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.41 - 3.70i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.14 + 2.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.66 + 2.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.62 + 6.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.67 + 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.19 - 3.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.52 + 4.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.48 - 9.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.70iT - 83T^{2} \)
89 \( 1 + (-6.10 - 3.52i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.12 - 4.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.129429468512931666614870112059, −8.027166411622826447500457335280, −7.31744210328809536777630221016, −6.55983744521427456394416388076, −6.19836393752161821366092299321, −5.10022772545416623255433090714, −3.94268849020671384892523421504, −3.40097958188640066301034354499, −2.25025369873161609108147832694, −1.24030672668590158831006939303, 0.828133369157619048612547560271, 1.79406449412410976103469972247, 2.91603405591717796267047030212, 3.92203455467833605760282758309, 5.11065840543143114149603330675, 5.62516987446994814672939322726, 5.98879621169831803004485897964, 7.28612171002848307920178834799, 8.176165936081006171233679156400, 8.958042135150556235988047922954

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