Properties

Label 2-2736-1.1-c1-0-5
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  + 1.59·5-s − 4.94·7-s − 5.59·11-s − 3.49·13-s + 7.09·17-s + 19-s + 3.19·23-s − 2.44·25-s − 4.69·29-s + 6·31-s − 7.89·35-s + 4·37-s + 2.50·41-s + 8.94·43-s + 6.28·47-s + 17.4·49-s + 5.69·53-s − 8.94·55-s + 6.99·59-s − 3.44·61-s − 5.58·65-s + 13.3·67-s − 13.3·71-s − 2.44·73-s + 27.6·77-s + 9.49·79-s + 0.691·83-s + ⋯
L(s)  = 1  + 0.714·5-s − 1.86·7-s − 1.68·11-s − 0.969·13-s + 1.72·17-s + 0.229·19-s + 0.666·23-s − 0.489·25-s − 0.871·29-s + 1.07·31-s − 1.33·35-s + 0.657·37-s + 0.390·41-s + 1.36·43-s + 0.917·47-s + 2.49·49-s + 0.782·53-s − 1.20·55-s + 0.910·59-s − 0.441·61-s − 0.692·65-s + 1.63·67-s − 1.58·71-s − 0.285·73-s + 3.15·77-s + 1.06·79-s + 0.0759·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258909162\)
\(L(\frac12)\) \(\approx\) \(1.258909162\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 + 4.94T + 7T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
23 \( 1 - 3.19T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 6.28T + 47T^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 - 6.99T + 59T^{2} \)
61 \( 1 + 3.44T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 2.44T + 73T^{2} \)
79 \( 1 - 9.49T + 79T^{2} \)
83 \( 1 - 0.691T + 83T^{2} \)
89 \( 1 - 8.69T + 89T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−9.135552802347914214544658739845, −7.80216187180713824272078678956, −7.43981927524138002216424981819, −6.44635628110503708359749117748, −5.63866573038222832661748356988, −5.28361910548045623745760877140, −3.90028888095897396760909962970, −2.86735129804313463807383461774, −2.49590539940599076898388139371, −0.66389323873369476547373894364, 0.66389323873369476547373894364, 2.49590539940599076898388139371, 2.86735129804313463807383461774, 3.90028888095897396760909962970, 5.28361910548045623745760877140, 5.63866573038222832661748356988, 6.44635628110503708359749117748, 7.43981927524138002216424981819, 7.80216187180713824272078678956, 9.135552802347914214544658739845

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