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Properties

Label 2-2736-76.27-c1-0-39
Degree $2$
Conductor $2736$
Sign $-0.536 + 0.844i$
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  + (−1.38 + 2.39i)5-s − 3.26i·7-s + 1.94i·11-s + (3.96 − 2.29i)13-s + (2.27 − 3.94i)17-s + (−3.43 − 2.68i)19-s + (−7.81 + 4.51i)23-s + (−1.32 − 2.29i)25-s + (−1.78 + 1.03i)29-s − 1.21·31-s + (7.81 + 4.51i)35-s + 1.73i·37-s + (−3.25 − 1.87i)41-s + (−1.82 − 1.05i)43-s + (−9.49 + 5.48i)47-s + ⋯
L(s)  = 1  + (−0.618 + 1.07i)5-s − 1.23i·7-s + 0.585i·11-s + (1.10 − 0.635i)13-s + (0.551 − 0.955i)17-s + (−0.787 − 0.615i)19-s + (−1.62 + 0.940i)23-s + (−0.264 − 0.458i)25-s + (−0.331 + 0.191i)29-s − 0.218·31-s + (1.32 + 0.762i)35-s + 0.284i·37-s + (−0.508 − 0.293i)41-s + (−0.278 − 0.160i)43-s + (−1.38 + 0.799i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.536 + 0.844i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.536 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6773907189\)
\(L(\frac12)\) \(\approx\) \(0.6773907189\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.43 + 2.68i)T \)
good5 \( 1 + (1.38 - 2.39i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.26iT - 7T^{2} \)
11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 + (-3.96 + 2.29i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.27 + 3.94i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (7.81 - 4.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.78 - 1.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + (3.25 + 1.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.82 + 1.05i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.49 - 5.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.40 + 4.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.322 - 0.559i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.04 + 7.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.13 + 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.04 + 8.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + (5.93 - 3.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.4 + 9.47i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.302692261042423669248608756828, −7.70881122877990042205125885037, −7.09566537935068464458148030843, −6.52143309482853182029535360491, −5.51303498735105662849593217232, −4.40127304281539239958146456517, −3.67034218781312364929959661130, −3.07108589997421157114102952026, −1.68284421503894606860011351721, −0.22418758474774317467018916686, 1.34401879822875435190137566324, 2.32770347409776957611614214300, 3.75867686082746229184408868814, 4.15619475735704231307929675761, 5.34034902463456459007103504714, 5.94455244341114495780540379762, 6.56985942670248214946772245290, 8.039682856791424107604131737882, 8.440092268077324261389139823469, 8.698380776436978674301791877518

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