Properties

Label 2-2736-1.1-c1-0-9
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  − 3.22·5-s + 2.37·7-s + 2.20·11-s + 2·13-s − 3.22·17-s − 19-s − 1.01·23-s + 5.37·25-s + 1.01·29-s − 4.74·31-s − 7.64·35-s + 10.7·37-s − 5.43·41-s + 11.1·43-s + 4.23·47-s − 1.37·49-s + 9.84·53-s − 7.11·55-s − 10.8·59-s − 5.11·61-s − 6.44·65-s + 4·67-s − 2.02·71-s − 5.11·73-s + 5.24·77-s + 4·79-s + 11.8·83-s + ⋯
L(s)  = 1  − 1.44·5-s + 0.896·7-s + 0.666·11-s + 0.554·13-s − 0.781·17-s − 0.229·19-s − 0.210·23-s + 1.07·25-s + 0.187·29-s − 0.852·31-s − 1.29·35-s + 1.76·37-s − 0.848·41-s + 1.69·43-s + 0.617·47-s − 0.196·49-s + 1.35·53-s − 0.959·55-s − 1.41·59-s − 0.655·61-s − 0.798·65-s + 0.488·67-s − 0.239·71-s − 0.598·73-s + 0.597·77-s + 0.450·79-s + 1.30·83-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

\(L(1)\) \(\approx\) \(1.483444259\)
\(L(\frac12)\) \(\approx\) \(1.483444259\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 3.22T + 17T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 5.43T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 4.23T + 47T^{2} \)
53 \( 1 - 9.84T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 2.02T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 - 9.84T + 89T^{2} \)
97 \( 1 - 7.48T + 97T^{2} \)
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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

−8.769359712107632604034576610804, −7.958604350175653133854217056348, −7.53751750734148443060072564470, −6.63663558871069114094398036591, −5.77308214461059582812985593409, −4.56426651941744798108376417970, −4.20762262891916088886630148617, −3.33642056655243582922104858778, −2.04243174127748300706522723755, −0.77047052557336047411496168979, 0.77047052557336047411496168979, 2.04243174127748300706522723755, 3.33642056655243582922104858778, 4.20762262891916088886630148617, 4.56426651941744798108376417970, 5.77308214461059582812985593409, 6.63663558871069114094398036591, 7.53751750734148443060072564470, 7.958604350175653133854217056348, 8.769359712107632604034576610804

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