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Properties

Label 2-2736-19.7-c1-0-3
Degree $2$
Conductor $2736$
Sign $-0.964 + 0.263i$
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 − 2.32·7-s − 1.70·11-s + (−2.01 + 3.48i)13-s + (−0.193 + 4.35i)19-s + (1.17 − 2.03i)23-s + (−3.01 + 5.22i)25-s + (3.32 − 5.75i)29-s − 6.70·31-s + (−3.85 − 6.67i)35-s − 37-s + (−3.32 − 5.75i)41-s + (0.353 + 0.612i)43-s + (3 − 5.19i)47-s − 1.61·49-s + ⋯
L(s)  = 1  + (0.742 + 1.28i)5-s − 0.877·7-s − 0.514·11-s + (−0.558 + 0.967i)13-s + (−0.0443 + 0.999i)19-s + (0.244 − 0.424i)23-s + (−0.602 + 1.04i)25-s + (0.616 − 1.06i)29-s − 1.20·31-s + (−0.651 − 1.12i)35-s − 0.164·37-s + (−0.518 − 0.898i)41-s + (0.0539 + 0.0934i)43-s + (0.437 − 0.757i)47-s − 0.230·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.964 + 0.263i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5487550262\)
\(L(\frac12)\) \(\approx\) \(0.5487550262\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (2.01 - 3.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.17 + 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.32 + 5.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (3.32 + 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.353 - 0.612i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.98 - 8.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.853 + 1.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.69 - 2.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.18 - 7.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.70 - 8.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.67 + 2.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (1.33 - 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.86 + 15.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.417669481621027151513885234542, −8.566000663654990794940681227938, −7.42425879171264075781425612686, −6.97093534771182888983278128057, −6.18310792152213182994670578116, −5.67765612549069325930901727930, −4.45202662315622975132039704083, −3.45290698426526208624439960836, −2.65951122310780824906612709034, −1.88451397674878543224435337605, 0.16713998856345718772603329228, 1.37783453169289343612390786265, 2.61666538947727443858617402679, 3.43716241866183184997895035283, 4.82720346680231829109173373973, 5.16011594590374936385166845908, 5.98370804802411753095379014859, 6.85932232905582628178199909535, 7.72108863396445674977483234755, 8.565557486848183820950380302047

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