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Properties

Label 2-2736-228.11-c1-0-27
Degree $2$
Conductor $2736$
Sign $0.986 + 0.164i$
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  + (3.37 + 1.95i)5-s − 3.27i·7-s + 5.67·11-s + (−2.34 − 4.06i)13-s + (1.20 + 0.695i)17-s + (3.88 + 1.97i)19-s + (3.59 + 6.22i)23-s + (5.10 + 8.84i)25-s + (−4.52 + 2.61i)29-s − 9.36i·31-s + (6.38 − 11.0i)35-s − 8.48·37-s + (3.32 + 1.91i)41-s + (0.584 + 0.337i)43-s + (−3.54 − 6.14i)47-s + ⋯
L(s)  = 1  + (1.51 + 0.872i)5-s − 1.23i·7-s + 1.71·11-s + (−0.651 − 1.12i)13-s + (0.292 + 0.168i)17-s + (0.891 + 0.453i)19-s + (0.749 + 1.29i)23-s + (1.02 + 1.76i)25-s + (−0.840 + 0.485i)29-s − 1.68i·31-s + (1.07 − 1.87i)35-s − 1.39·37-s + (0.519 + 0.299i)41-s + (0.0891 + 0.0514i)43-s + (−0.517 − 0.896i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.798552807\)
\(L(\frac12)\) \(\approx\) \(2.798552807\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.88 - 1.97i)T \)
good5 \( 1 + (-3.37 - 1.95i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.27iT - 7T^{2} \)
11 \( 1 - 5.67T + 11T^{2} \)
13 \( 1 + (2.34 + 4.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.20 - 0.695i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.59 - 6.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.52 - 2.61i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.36iT - 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + (-3.32 - 1.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.584 - 0.337i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.54 + 6.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.58 - 2.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.63 + 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.574 - 0.994i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.96 - 3.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.92 + 8.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.95 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.25 - 3.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.05T + 83T^{2} \)
89 \( 1 + (13.8 - 7.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.66 + 8.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.150221595426406984744340995767, −7.76566557771795651649399567593, −7.22210453664471027885150218217, −6.56447792926413468166386880754, −5.77744851931955627176568218759, −5.11426706384629517078398346218, −3.72687806950797136919028704075, −3.27545534754560205519910284483, −1.93435699110504394229914611888, −1.08228155698867202687976630056, 1.25118347452738853308991660907, 1.97731478027347002249199616426, 2.91301200255037753978571632866, 4.31016683305410564834531542330, 5.10772043476012822747378829636, 5.66005451091122348306917997300, 6.55346817760817694507522095192, 7.00339243135192497493667681696, 8.589294637335677155078954499594, 8.950631889105082789544636009444

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