Properties

Label 2-2736-3.2-c2-0-31
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  + 5.71i·5-s − 5.91·7-s + 2.41i·11-s + 17.1·13-s − 16.1i·17-s − 4.35·19-s − 15.9i·23-s − 7.70·25-s + 31.2i·29-s + 13.5·31-s − 33.8i·35-s + 45.7·37-s + 30.3i·41-s + 21.0·43-s − 43.6i·47-s + ⋯
L(s)  = 1  + 1.14i·5-s − 0.845·7-s + 0.219i·11-s + 1.31·13-s − 0.949i·17-s − 0.229·19-s − 0.694i·23-s − 0.308·25-s + 1.07i·29-s + 0.435·31-s − 0.966i·35-s + 1.23·37-s + 0.740i·41-s + 0.490·43-s − 0.928i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.909792384\)
\(L(\frac12)\) \(\approx\) \(1.909792384\)
\(L(2)\) 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 \( 1 - 5.71iT - 25T^{2} \)
7 \( 1 + 5.91T + 49T^{2} \)
11 \( 1 - 2.41iT - 121T^{2} \)
13 \( 1 - 17.1T + 169T^{2} \)
17 \( 1 + 16.1iT - 289T^{2} \)
23 \( 1 + 15.9iT - 529T^{2} \)
29 \( 1 - 31.2iT - 841T^{2} \)
31 \( 1 - 13.5T + 961T^{2} \)
37 \( 1 - 45.7T + 1.36e3T^{2} \)
41 \( 1 - 30.3iT - 1.68e3T^{2} \)
43 \( 1 - 21.0T + 1.84e3T^{2} \)
47 \( 1 + 43.6iT - 2.20e3T^{2} \)
53 \( 1 + 43.5iT - 2.80e3T^{2} \)
59 \( 1 + 86.2iT - 3.48e3T^{2} \)
61 \( 1 - 112.T + 3.72e3T^{2} \)
67 \( 1 - 41.3T + 4.48e3T^{2} \)
71 \( 1 + 13.6iT - 5.04e3T^{2} \)
73 \( 1 + 21.7T + 5.32e3T^{2} \)
79 \( 1 + 71.2T + 6.24e3T^{2} \)
83 \( 1 - 19.9iT - 6.88e3T^{2} \)
89 \( 1 - 20.3iT - 7.92e3T^{2} \)
97 \( 1 - 20.1T + 9.40e3T^{2} \)
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

−8.752576527692228491183072279612, −8.015330566016224323962319163590, −6.91658160766410928144776389387, −6.67049672047000848787504261290, −5.91204019006235575173885202084, −4.84879343347608185279586714272, −3.73926694402559367294133655571, −3.11865558071565396404131421603, −2.29263458020024365362627951130, −0.802353819745480361231943411805, 0.63287509493955731255891897022, 1.51300042770659288329828796182, 2.84289634278180827298597191337, 3.90350850344438932354827495325, 4.39917885874753931996186538597, 5.74864020899212129521327473298, 5.94975666781687744526406280667, 6.94957503956220958935704101613, 8.056635334255227045032620629752, 8.500426966727475682033820876046

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