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Properties

Label 2-2736-57.56-c1-0-1
Degree $2$
Conductor $2736$
Sign $-0.577 + 0.816i$
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  + 3.95i·5-s − 4.77·7-s + 5.95i·11-s + 5.63i·17-s − 4.35·19-s − 3.33i·23-s − 10.6·25-s − 18.8i·35-s + 10.8·43-s − 0.845i·47-s + 15.8·49-s − 23.5·55-s + 15.1·61-s − 16.8·73-s − 28.4i·77-s + ⋯
L(s)  = 1  + 1.76i·5-s − 1.80·7-s + 1.79i·11-s + 1.36i·17-s − 1.00·19-s − 0.695i·23-s − 2.12·25-s − 3.19i·35-s + 1.65·43-s − 0.123i·47-s + 2.26·49-s − 3.17·55-s + 1.94·61-s − 1.96·73-s − 3.24i·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5666773354\)
\(L(\frac12)\) \(\approx\) \(0.5666773354\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35T \)
good5 \( 1 - 3.95iT - 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 - 5.95iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.63iT - 17T^{2} \)
23 \( 1 + 3.33iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 0.845iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 5.14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.594343286569441468524558964257, −8.611326643463051349319724723000, −7.44680919032196651596862446466, −6.96519535629429471758075008788, −6.40024196163116676268577008519, −5.88049426447121337754709351152, −4.28239351266222366617840654647, −3.70966363281998011142306062997, −2.71806484957337314713512538682, −2.12223927202963058311630871797, 0.21922955850372402693366741329, 0.937823593254205181878926434019, 2.60928584283827658675864979218, 3.51935319289925344322605803965, 4.29459447822260081171838121278, 5.41559972546552638451734856441, 5.85672865973356957189600407017, 6.67217952949657642323863592554, 7.69460769003999616344568572296, 8.611025506801124206012052837427

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