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Properties

Label 2-2736-57.8-c1-0-15
Degree $2$
Conductor $2736$
Sign $0.903 + 0.429i$
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  + (−3.12 − 1.80i)5-s + 3.25·7-s − 2.41i·11-s + (−3.29 + 1.90i)13-s + (5.21 + 3.00i)17-s + (1.75 + 3.99i)19-s + (1.58 − 0.915i)23-s + (4.00 + 6.93i)25-s + (−1.46 − 2.54i)29-s + 1.11i·31-s + (−10.1 − 5.86i)35-s + 10.5i·37-s + (0.433 − 0.749i)41-s + (0.875 − 1.51i)43-s + (5.51 − 3.18i)47-s + ⋯
L(s)  = 1  + (−1.39 − 0.806i)5-s + 1.23·7-s − 0.727i·11-s + (−0.914 + 0.528i)13-s + (1.26 + 0.729i)17-s + (0.401 + 0.915i)19-s + (0.330 − 0.190i)23-s + (0.800 + 1.38i)25-s + (−0.272 − 0.472i)29-s + 0.201i·31-s + (−1.71 − 0.992i)35-s + 1.73i·37-s + (0.0676 − 0.117i)41-s + (0.133 − 0.231i)43-s + (0.804 − 0.464i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.903 + 0.429i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.903 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.534810636\)
\(L(\frac12)\) \(\approx\) \(1.534810636\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 - 3.99i)T \)
good5 \( 1 + (3.12 + 1.80i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 + (3.29 - 1.90i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.21 - 3.00i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.58 + 0.915i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.46 + 2.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.11iT - 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 + (-0.433 + 0.749i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.875 + 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.51 + 3.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.08 + 3.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.55 + 6.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.59 + 6.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.502 + 0.869i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.05 + 12.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.03 - 1.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.8iT - 83T^{2} \)
89 \( 1 + (-2.62 - 4.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.737 + 0.425i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.446595774381826717587163531569, −7.992727996097691659616989489634, −7.69013398619049953966647965565, −6.56122394119802278112608645161, −5.35110421981362863491838637280, −4.91023340006512600145041530420, −4.00974468686033710730929574610, −3.33602383428745922042589192868, −1.80038987382010449481743592560, −0.76941398849144673047437918347, 0.811870812151986389914248621657, 2.36146375951734178873219204195, 3.18032015134687380789634857696, 4.16982746615029891345574362036, 4.89608366822591105378962648773, 5.59420757517184577497821931267, 7.10603855791187504884227208123, 7.42017820677623700950386817219, 7.77915738179850058917281969898, 8.753403820966418726409576946616

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