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Properties

Label 2-2736-228.11-c1-0-21
Degree $2$
Conductor $2736$
Sign $0.950 - 0.310i$
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  + (3.43 + 1.98i)5-s − 2.01i·7-s − 3.45·11-s + (0.581 + 1.00i)13-s + (−1.94 − 1.12i)17-s + (4.27 − 0.843i)19-s + (−1.36 − 2.36i)23-s + (5.36 + 9.28i)25-s + (−0.0525 + 0.0303i)29-s − 2.92i·31-s + (3.99 − 6.91i)35-s + 10.5·37-s + (9.71 + 5.60i)41-s + (6.21 + 3.58i)43-s + (5.99 + 10.3i)47-s + ⋯
L(s)  = 1  + (1.53 + 0.886i)5-s − 0.761i·7-s − 1.04·11-s + (0.161 + 0.279i)13-s + (−0.470 − 0.271i)17-s + (0.981 − 0.193i)19-s + (−0.284 − 0.493i)23-s + (1.07 + 1.85i)25-s + (−0.00976 + 0.00563i)29-s − 0.525i·31-s + (0.674 − 1.16i)35-s + 1.73·37-s + (1.51 + 0.875i)41-s + (0.947 + 0.547i)43-s + (0.873 + 1.51i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.950 - 0.310i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.950 - 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.427169250\)
\(L(\frac12)\) \(\approx\) \(2.427169250\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.27 + 0.843i)T \)
good5 \( 1 + (-3.43 - 1.98i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.01iT - 7T^{2} \)
11 \( 1 + 3.45T + 11T^{2} \)
13 \( 1 + (-0.581 - 1.00i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.94 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.36 + 2.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0525 - 0.0303i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.92iT - 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + (-9.71 - 5.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.21 - 3.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.99 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.31 - 1.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.08 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.76 + 6.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 - 1.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.20 - 7.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.71 + 2.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.16 - 2.98i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (12.2 - 7.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.03 + 8.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.191043544987429078937929501937, −7.83312874526606000877878960460, −7.41113314085833871014961846805, −6.39928092086549374622673204572, −5.98714372892881782667144952341, −5.05859326067525536691424040970, −4.14072258363860134540690625396, −2.80922726030183751757795824501, −2.41428471532576777013433816305, −1.04851649998740398025624158040, 0.963630043890834617379004725404, 2.15038602610990110361436463653, 2.71296504261038851990468407850, 4.15760662423796672466805016602, 5.24998242488257955368703875511, 5.59630217481616268202663681686, 6.14202923216157879894572859940, 7.36133964351731544463291256556, 8.142043055689838217229774551717, 9.126138009345370725133791433584

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