Properties

Label 2-2736-19.18-c2-0-98
Degree $2$
Conductor $2736$
Sign $-1$
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  − 4·5-s + 10·7-s + 10·11-s − 24.2i·13-s − 10·17-s − 19·19-s − 20·23-s − 9·25-s + 34.6i·29-s − 17.3i·31-s − 40·35-s − 10.3i·37-s − 34.6i·41-s + 10·43-s − 80·47-s + ⋯
L(s)  = 1  − 0.800·5-s + 1.42·7-s + 0.909·11-s − 1.86i·13-s − 0.588·17-s − 19-s − 0.869·23-s − 0.359·25-s + 1.19i·29-s − 0.558i·31-s − 1.14·35-s − 0.280i·37-s − 0.844i·41-s + 0.232·43-s − 1.70·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3731935293\)
\(L(\frac12)\) \(\approx\) \(0.3731935293\)
\(L(2)\) 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 \( 1 + 19T \)
good5 \( 1 + 4T + 25T^{2} \)
7 \( 1 - 10T + 49T^{2} \)
11 \( 1 - 10T + 121T^{2} \)
13 \( 1 + 24.2iT - 169T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
23 \( 1 + 20T + 529T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 + 17.3iT - 961T^{2} \)
37 \( 1 + 10.3iT - 1.36e3T^{2} \)
41 \( 1 + 34.6iT - 1.68e3T^{2} \)
43 \( 1 - 10T + 1.84e3T^{2} \)
47 \( 1 + 80T + 2.20e3T^{2} \)
53 \( 1 + 41.5iT - 2.80e3T^{2} \)
59 \( 1 - 34.6iT - 3.48e3T^{2} \)
61 \( 1 + 10T + 3.72e3T^{2} \)
67 \( 1 - 76.2iT - 4.48e3T^{2} \)
71 \( 1 - 103. iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 - 17.3iT - 6.24e3T^{2} \)
83 \( 1 - 70T + 6.88e3T^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 - 76.2iT - 9.40e3T^{2} \)
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

−8.275894058089597159868054884042, −7.70419885108029885613329107263, −6.88093828368026515409070534189, −5.85803290473776230659147779265, −5.09827804762983624473892201321, −4.23558673262339735914459964559, −3.61096462915241623382695676982, −2.34942641025673812072461627161, −1.29889896153095985114723183451, −0.085063885152692211467309290560, 1.54385997192990941027552130606, 2.11347530713007540101659571774, 3.71485947343092483477021546398, 4.42214055269399308662618963456, 4.71596957154463816108555574343, 6.21384291701002508986663312584, 6.65767758222565868448497478835, 7.71532754222860423244608047131, 8.182192272491092724713868251433, 8.939740459607827378890608954137

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