Properties

Label 2-2736-76.27-c1-0-22
Degree $2$
Conductor $2736$
Sign $0.865 - 0.500i$
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  + (−2.00 + 3.47i)5-s − 1.92i·7-s − 1.38i·11-s + (0.470 − 0.271i)13-s + (−1.83 + 3.17i)17-s + (−1.80 − 3.96i)19-s + (7.04 − 4.06i)23-s + (−5.53 − 9.59i)25-s + (2.40 − 1.38i)29-s + 7.06·31-s + (6.70 + 3.86i)35-s + 2.12i·37-s + (8.24 + 4.75i)41-s + (−5.88 − 3.39i)43-s + (−6.18 + 3.57i)47-s + ⋯
L(s)  = 1  + (−0.896 + 1.55i)5-s − 0.729i·7-s − 0.417i·11-s + (0.130 − 0.0753i)13-s + (−0.444 + 0.770i)17-s + (−0.413 − 0.910i)19-s + (1.46 − 0.847i)23-s + (−1.10 − 1.91i)25-s + (0.445 − 0.257i)29-s + 1.26·31-s + (1.13 + 0.653i)35-s + 0.349i·37-s + (1.28 + 0.743i)41-s + (−0.897 − 0.518i)43-s + (−0.902 + 0.520i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.398166402\)
\(L(\frac12)\) \(\approx\) \(1.398166402\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 3.96i)T \)
good5 \( 1 + (2.00 - 3.47i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.92iT - 7T^{2} \)
11 \( 1 + 1.38iT - 11T^{2} \)
13 \( 1 + (-0.470 + 0.271i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.83 - 3.17i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.04 + 4.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.40 + 1.38i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.06T + 31T^{2} \)
37 \( 1 - 2.12iT - 37T^{2} \)
41 \( 1 + (-8.24 - 4.75i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.88 + 3.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.18 - 3.57i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.171 - 0.0988i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.80 - 4.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.871 - 1.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.34 - 12.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.17 + 3.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.50 + 6.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.54 + 7.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (15.3 - 8.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.97 - 4.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.643717859436515516767429591438, −8.112924770635903964503337830182, −7.22339159335028310294887833071, −6.69797512873508348877096324287, −6.18607021468123506966608507311, −4.71834134256808093426828581758, −4.07319368087223627114331267106, −3.14725346038683495023193949011, −2.54242698626176896711409172790, −0.75853069201616331053007131459, 0.71772165287617978605252561150, 1.84988152243621682035901465584, 3.14273317619728373257676759588, 4.15548188128013688079457703137, 4.89228123597210749649732355359, 5.38805963365072951241183533895, 6.47497013100582118559702693038, 7.43410794691853889694435280183, 8.138567236342910198416883421111, 8.769985256107644254273860730923

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