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Properties

Label 2-2736-228.11-c1-0-19
Degree $2$
Conductor $2736$
Sign $0.871 - 0.490i$
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  + (2.27 + 1.31i)5-s − 3.01i·7-s − 0.730·11-s + (2.20 + 3.81i)13-s + (5.35 + 3.08i)17-s + (−2.91 − 3.24i)19-s + (2.23 + 3.86i)23-s + (0.957 + 1.65i)25-s + (−4.85 + 2.80i)29-s + 6.37i·31-s + (3.96 − 6.87i)35-s − 1.48·37-s + (3.54 + 2.04i)41-s + (6.82 + 3.93i)43-s + (−1.28 − 2.22i)47-s + ⋯
L(s)  = 1  + (1.01 + 0.587i)5-s − 1.14i·7-s − 0.220·11-s + (0.610 + 1.05i)13-s + (1.29 + 0.749i)17-s + (−0.667 − 0.744i)19-s + (0.465 + 0.806i)23-s + (0.191 + 0.331i)25-s + (−0.900 + 0.520i)29-s + 1.14i·31-s + (0.670 − 1.16i)35-s − 0.244·37-s + (0.554 + 0.319i)41-s + (1.04 + 0.600i)43-s + (−0.187 − 0.324i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.871 - 0.490i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.316279332\)
\(L(\frac12)\) \(\approx\) \(2.316279332\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 + 3.24i)T \)
good5 \( 1 + (-2.27 - 1.31i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.01iT - 7T^{2} \)
11 \( 1 + 0.730T + 11T^{2} \)
13 \( 1 + (-2.20 - 3.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.35 - 3.08i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.23 - 3.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.85 - 2.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.37iT - 31T^{2} \)
37 \( 1 + 1.48T + 37T^{2} \)
41 \( 1 + (-3.54 - 2.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.82 - 3.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.28 + 2.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.22 + 5.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 - 1.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.26 + 7.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.40 + 4.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.00 + 8.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.72 - 9.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.78 + 1.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.34T + 83T^{2} \)
89 \( 1 + (-2.22 + 1.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.80 - 4.86i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.059574628619875501555774305092, −8.057530064183567746941216014729, −7.20838886843901293441217542904, −6.65669018295334569491814245207, −5.92081152838223802512287516149, −5.05883875400290459171479984571, −3.99307424991011382089560829786, −3.30017910907340595177229234716, −2.07161385301023208809061740629, −1.17172191156904960034124341392, 0.859820460378700141638955595031, 2.09960119428339931422178109610, 2.82685985354718265089583884907, 3.98696709466372685381026359696, 5.19551498696403612540388893028, 5.76868454536937235392779548950, 5.99353595228596476528540023429, 7.38098442720527137480443749220, 8.126270985350824924887294942714, 8.854329550658131691532875928162

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