Properties

Label 2-2736-19.18-c2-0-4
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  − 7·5-s − 11·7-s + 3·11-s + 11.3i·13-s + 17·17-s + 19·19-s + 2·23-s + 24·25-s − 39.5i·29-s + 5.65i·31-s + 77·35-s + 39.5i·37-s + 39.5i·41-s + 21·43-s − 5·47-s + ⋯
L(s)  = 1  − 1.40·5-s − 1.57·7-s + 0.272·11-s + 0.870i·13-s + 17-s + 19-s + 0.0869·23-s + 0.959·25-s − 1.36i·29-s + 0.182i·31-s + 2.19·35-s + 1.07i·37-s + 0.965i·41-s + 0.488·43-s − 0.106·47-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

\(L(\frac{3}{2})\) \(\approx\) \(0.1630138767\)
\(L(\frac12)\) \(\approx\) \(0.1630138767\)
\(L(2)\) 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 \( 1 + 7T + 25T^{2} \)
7 \( 1 + 11T + 49T^{2} \)
11 \( 1 - 3T + 121T^{2} \)
13 \( 1 - 11.3iT - 169T^{2} \)
17 \( 1 - 17T + 289T^{2} \)
23 \( 1 - 2T + 529T^{2} \)
29 \( 1 + 39.5iT - 841T^{2} \)
31 \( 1 - 5.65iT - 961T^{2} \)
37 \( 1 - 39.5iT - 1.36e3T^{2} \)
41 \( 1 - 39.5iT - 1.68e3T^{2} \)
43 \( 1 - 21T + 1.84e3T^{2} \)
47 \( 1 + 5T + 2.20e3T^{2} \)
53 \( 1 - 5.65iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 - 23T + 3.72e3T^{2} \)
67 \( 1 - 39.5iT - 4.48e3T^{2} \)
71 \( 1 + 90.5iT - 5.04e3T^{2} \)
73 \( 1 - 39T + 5.32e3T^{2} \)
79 \( 1 + 96.1iT - 6.24e3T^{2} \)
83 \( 1 + 6T + 6.88e3T^{2} \)
89 \( 1 - 118. iT - 7.92e3T^{2} \)
97 \( 1 - 169. iT - 9.40e3T^{2} \)
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

−9.177305583376216898726970618883, −8.081135826890479297695974816702, −7.59151813140525724586077676305, −6.72442401389623998453843864096, −6.20141626398805930106019693827, −5.05983668902382265487330444527, −4.03188017027254194821607328455, −3.51711908111293772617283926915, −2.73915680265906315362163159474, −1.05039645358483421031747458107, 0.05492748429264239096485275246, 0.967490205696825415777589730866, 2.83421843604645816279824663538, 3.45799986699934229368587296129, 3.96103636500002537358321257262, 5.24965394737118642965173912021, 5.91529865222587186567521362696, 7.04680744191219891712001137407, 7.35407086765823846071304169297, 8.208850659089930954135162480977

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