Properties

Label 2-2736-1.1-c1-0-0
Degree $2$
Conductor $2736$
Sign $1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15·5-s − 3.37·7-s − 4.70·11-s + 2·13-s − 2.15·17-s − 19-s − 6.85·23-s − 0.372·25-s + 6.85·29-s + 6.74·31-s + 7.25·35-s − 0.744·37-s + 2.55·41-s − 6.11·43-s + 9.00·47-s + 4.37·49-s − 11.9·53-s + 10.1·55-s + 5.10·59-s + 12.1·61-s − 4.30·65-s + 4·67-s − 13.7·71-s + 12.1·73-s + 15.8·77-s + 4·79-s + 1.75·83-s + ⋯
L(s)  = 1  − 0.962·5-s − 1.27·7-s − 1.41·11-s + 0.554·13-s − 0.521·17-s − 0.229·19-s − 1.42·23-s − 0.0744·25-s + 1.27·29-s + 1.21·31-s + 1.22·35-s − 0.122·37-s + 0.398·41-s − 0.932·43-s + 1.31·47-s + 0.624·49-s − 1.64·53-s + 1.36·55-s + 0.664·59-s + 1.55·61-s − 0.533·65-s + 0.488·67-s − 1.62·71-s + 1.41·73-s + 1.80·77-s + 0.450·79-s + 0.192·83-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.6718268525\)
\(L(\frac12)\) \(\approx\) \(0.6718268525\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 + 2.15T + 5T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
23 \( 1 + 6.85T + 23T^{2} \)
29 \( 1 - 6.85T + 29T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 + 0.744T + 37T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 + 6.11T + 43T^{2} \)
47 \( 1 - 9.00T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 - 5.10T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−8.542703413663872604287355756967, −8.188085705404294572356566101304, −7.36396442346700714759525199173, −6.49794453748563139172271638055, −5.91481280872669157378273828487, −4.79611241091220360371901940495, −3.97823690290518255640872029998, −3.17966453059115101880052687183, −2.33466469563871227316883924625, −0.47879729343034893960514321785, 0.47879729343034893960514321785, 2.33466469563871227316883924625, 3.17966453059115101880052687183, 3.97823690290518255640872029998, 4.79611241091220360371901940495, 5.91481280872669157378273828487, 6.49794453748563139172271638055, 7.36396442346700714759525199173, 8.188085705404294572356566101304, 8.542703413663872604287355756967

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