Properties

Label 2-72e2-1.1-c1-0-1
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  − 3.37·5-s − 1.37·7-s + 11-s − 5.37·13-s − 0.372·17-s − 6.37·19-s − 5.37·23-s + 6.37·25-s + 1.37·29-s + 0.627·31-s + 4.62·35-s + 2.74·37-s + 0.255·41-s − 9.74·43-s + 1.37·47-s − 5.11·49-s − 10.7·53-s − 3.37·55-s + 7·59-s + 3.37·61-s + 18.1·65-s − 7.74·67-s + 4·71-s + 5.11·73-s − 1.37·77-s − 0.627·79-s + 15.3·83-s + ⋯
L(s)  = 1  − 1.50·5-s − 0.518·7-s + 0.301·11-s − 1.49·13-s − 0.0902·17-s − 1.46·19-s − 1.12·23-s + 1.27·25-s + 0.254·29-s + 0.112·31-s + 0.782·35-s + 0.451·37-s + 0.0398·41-s − 1.48·43-s + 0.200·47-s − 0.730·49-s − 1.47·53-s − 0.454·55-s + 0.911·59-s + 0.431·61-s + 2.24·65-s − 0.946·67-s + 0.474·71-s + 0.598·73-s − 0.156·77-s − 0.0706·79-s + 1.68·83-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

\(L(1)\) \(\approx\) \(0.4242817668\)
\(L(\frac12)\) \(\approx\) \(0.4242817668\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.37T + 5T^{2} \)
7 \( 1 + 1.37T + 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + 5.37T + 13T^{2} \)
17 \( 1 + 0.372T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 - 0.627T + 31T^{2} \)
37 \( 1 - 2.74T + 37T^{2} \)
41 \( 1 - 0.255T + 41T^{2} \)
43 \( 1 + 9.74T + 43T^{2} \)
47 \( 1 - 1.37T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 + 7.74T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + 0.627T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 9.74T + 97T^{2} \)
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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

−8.084434864127599446233812559273, −7.62565201392848534208393950920, −6.76822183873627090848629986429, −6.31442473907196185258306332949, −5.08555538358555783927809843297, −4.40839239828497161791199787234, −3.81688924256423231085219439691, −2.95900043919272169232512714004, −1.99153042970297041116046491294, −0.33248652120329347485811586158, 0.33248652120329347485811586158, 1.99153042970297041116046491294, 2.95900043919272169232512714004, 3.81688924256423231085219439691, 4.40839239828497161791199787234, 5.08555538358555783927809843297, 6.31442473907196185258306332949, 6.76822183873627090848629986429, 7.62565201392848534208393950920, 8.084434864127599446233812559273

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