Properties

Label 2-2736-3.2-c2-0-21
Degree $2$
Conductor $2736$
Sign $-0.577 - 0.816i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.29i·5-s + 2.46·7-s + 10.4i·11-s + 2.93·13-s + 27.8i·17-s + 4.35·19-s − 24.3i·23-s + 14.1·25-s + 7.80i·29-s + 17.4·31-s + 8.11i·35-s − 48.3·37-s + 51.2i·41-s + 82.7·43-s + 22.2i·47-s + ⋯
L(s)  = 1  + 0.659i·5-s + 0.351·7-s + 0.946i·11-s + 0.225·13-s + 1.63i·17-s + 0.229·19-s − 1.05i·23-s + 0.565·25-s + 0.269i·29-s + 0.562·31-s + 0.231i·35-s − 1.30·37-s + 1.24i·41-s + 1.92·43-s + 0.472i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.780486314\)
\(L(\frac12)\) \(\approx\) \(1.780486314\)
\(L(2)\) 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 \( 1 - 3.29iT - 25T^{2} \)
7 \( 1 - 2.46T + 49T^{2} \)
11 \( 1 - 10.4iT - 121T^{2} \)
13 \( 1 - 2.93T + 169T^{2} \)
17 \( 1 - 27.8iT - 289T^{2} \)
23 \( 1 + 24.3iT - 529T^{2} \)
29 \( 1 - 7.80iT - 841T^{2} \)
31 \( 1 - 17.4T + 961T^{2} \)
37 \( 1 + 48.3T + 1.36e3T^{2} \)
41 \( 1 - 51.2iT - 1.68e3T^{2} \)
43 \( 1 - 82.7T + 1.84e3T^{2} \)
47 \( 1 - 22.2iT - 2.20e3T^{2} \)
53 \( 1 - 16.6iT - 2.80e3T^{2} \)
59 \( 1 + 49.5iT - 3.48e3T^{2} \)
61 \( 1 - 16.9T + 3.72e3T^{2} \)
67 \( 1 - 1.05T + 4.48e3T^{2} \)
71 \( 1 + 79.0iT - 5.04e3T^{2} \)
73 \( 1 + 53.4T + 5.32e3T^{2} \)
79 \( 1 + 137.T + 6.24e3T^{2} \)
83 \( 1 - 4.56iT - 6.88e3T^{2} \)
89 \( 1 - 120. iT - 7.92e3T^{2} \)
97 \( 1 - 13.4T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.808305193158162219345170729181, −8.150747362843536545792638706202, −7.38668975346161901153290623023, −6.59409324587649368894467056396, −6.02765167841203906511483982829, −4.89641188305264077382649163621, −4.22111419741005675068607742078, −3.23056894493819671521304726178, −2.25589110281936665105561478224, −1.28747003470351549491335247565, 0.43745811850339143720514683145, 1.33120192575877164837269212714, 2.63543127509638403146209030017, 3.52454116835914242439740401617, 4.54147376784671432730810861688, 5.31202975902303444637409665254, 5.86343970157321095850339897543, 7.04190831932895351534421579641, 7.57444252305955207012695023359, 8.632463550747653659190631117259

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