Properties

Label 2-72e2-1.1-c1-0-22
Degree $2$
Conductor $5184$
Sign $1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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  + 2.64·5-s − 0.913·7-s − 5.58·11-s − 2.58·13-s + 1.73·17-s + 7.84·19-s − 1.58·23-s + 2.00·25-s − 0.818·29-s − 1.82·31-s − 2.41·35-s + 6.58·37-s + 8.75·41-s − 7.84·43-s + 8·47-s − 6.16·49-s + 8.75·53-s − 14.7·55-s − 8·59-s + 10.5·61-s − 6.83·65-s − 7.84·67-s + 9.58·71-s + 12.1·73-s + 5.10·77-s + 14.7·79-s + 3.16·83-s + ⋯
L(s)  = 1  + 1.18·5-s − 0.345·7-s − 1.68·11-s − 0.716·13-s + 0.420·17-s + 1.79·19-s − 0.329·23-s + 0.400·25-s − 0.151·29-s − 0.328·31-s − 0.408·35-s + 1.08·37-s + 1.36·41-s − 1.19·43-s + 1.16·47-s − 0.880·49-s + 1.20·53-s − 1.99·55-s − 1.04·59-s + 1.35·61-s − 0.847·65-s − 0.958·67-s + 1.13·71-s + 1.42·73-s + 0.581·77-s + 1.66·79-s + 0.347·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.062119285\)
\(L(\frac12)\) \(\approx\) \(2.062119285\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 + 0.913T + 7T^{2} \)
11 \( 1 + 5.58T + 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 7.84T + 19T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 + 0.818T + 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 6.58T + 37T^{2} \)
41 \( 1 - 8.75T + 41T^{2} \)
43 \( 1 + 7.84T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 8.75T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 7.84T + 67T^{2} \)
71 \( 1 - 9.58T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 - 8.85T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.900098559370145985834436400506, −7.69755234905245999702693777280, −6.76080483126390297048877546725, −5.83083701135125561429572846504, −5.40969790255610630473074066901, −4.82710764614132006659288155018, −3.52600479177222217942188144372, −2.68382050932621687628790664760, −2.10256274500491379916328157803, −0.76233114315788981028630331725, 0.76233114315788981028630331725, 2.10256274500491379916328157803, 2.68382050932621687628790664760, 3.52600479177222217942188144372, 4.82710764614132006659288155018, 5.40969790255610630473074066901, 5.83083701135125561429572846504, 6.76080483126390297048877546725, 7.69755234905245999702693777280, 7.900098559370145985834436400506

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