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Properties

Label 2-2736-228.83-c1-0-18
Degree $2$
Conductor $2736$
Sign $0.909 - 0.416i$
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  + (−1.45 + 0.842i)5-s + 0.172i·7-s + 2.01·11-s + (−0.109 + 0.189i)13-s + (5.01 − 2.89i)17-s + (3.23 − 2.92i)19-s + (−4.27 + 7.40i)23-s + (−1.08 + 1.87i)25-s + (0.957 + 0.552i)29-s − 8.93i·31-s + (−0.145 − 0.251i)35-s − 6.80·37-s + (−5.97 + 3.44i)41-s + (4.91 − 2.83i)43-s + (1.15 − 1.99i)47-s + ⋯
L(s)  = 1  + (−0.652 + 0.376i)5-s + 0.0651i·7-s + 0.608·11-s + (−0.0302 + 0.0524i)13-s + (1.21 − 0.702i)17-s + (0.742 − 0.670i)19-s + (−0.891 + 1.54i)23-s + (−0.216 + 0.374i)25-s + (0.177 + 0.102i)29-s − 1.60i·31-s + (−0.0245 − 0.0424i)35-s − 1.11·37-s + (−0.932 + 0.538i)41-s + (0.748 − 0.432i)43-s + (0.168 − 0.291i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628493315\)
\(L(\frac12)\) \(\approx\) \(1.628493315\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 2.92i)T \)
good5 \( 1 + (1.45 - 0.842i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.172iT - 7T^{2} \)
11 \( 1 - 2.01T + 11T^{2} \)
13 \( 1 + (0.109 - 0.189i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.01 + 2.89i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.27 - 7.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.957 - 0.552i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.93iT - 31T^{2} \)
37 \( 1 + 6.80T + 37T^{2} \)
41 \( 1 + (5.97 - 3.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.91 + 2.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.15 + 1.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.55 + 2.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.14 - 8.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.39 + 9.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 6.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.30 - 12.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.47 + 4.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.20 + 3.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-0.701 - 0.404i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.990 + 1.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.956548363204275228133415049808, −7.87423160902909097783639143457, −7.48114605173302511998798827251, −6.74936078313779406421319786149, −5.69619037224999493124504698105, −5.09810659550953423666821714946, −3.80850695178260928312861706972, −3.46092846739795828411234624937, −2.21848550498819577321134491022, −0.880949341018472211041143278471, 0.74850431605458550642616937457, 1.90452979009008164289225988189, 3.31187427774720602754331283945, 3.90929822501557806253629152680, 4.80604707835718284719110445857, 5.68411064396115508650898511167, 6.50316771221588388495312877799, 7.31789255085311637789016057335, 8.249968384676266628355787862708, 8.459553605027933766189496883609

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