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Properties

Label 2-2736-228.11-c1-0-26
Degree $2$
Conductor $2736$
Sign $-0.946 + 0.322i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.946 + 0.322i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01435939591\)
\(L(\frac12)\) \(\approx\) \(0.01435939591\)
\(L(\frac{3}{2})\) 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

−8.498610362960854899235069378062, −7.908935575733257967094035271517, −6.98955946059913016779251656749, −6.18675437664902470570008625822, −5.27833698638689619368124317995, −4.53624847877848383168721314837, −3.75819931329030177692703898757, −2.68819096014380202032815492240, −1.60073144628422219921914914479, −0.00498791119319287142748689123, 1.24873287882811805633139166510, 2.92704215920781008972687657407, 3.46065437812795061530933809022, 4.40156343414228897741263756788, 5.07101038734492702568334627602, 6.46931147207111877784440445164, 6.83925680268378430362861231298, 7.68371957550250102881794453738, 8.263140707465476514636932113942, 8.979624510619028017912615853631

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