Properties

Label 2-72e2-1.1-c1-0-12
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  − 1.73·5-s + 2·7-s − 3.46·11-s + 13-s − 5.19·17-s − 2·19-s − 3.46·23-s − 2.00·25-s + 1.73·29-s + 8·31-s − 3.46·35-s + 7·37-s + 6.92·41-s − 2·43-s − 6.92·47-s − 3·49-s + 5.99·55-s + 13.8·59-s + 7·61-s − 1.73·65-s + 10·67-s + 10.3·71-s − 7·73-s − 6.92·77-s + 2·79-s − 13.8·83-s + 9·85-s + ⋯
L(s)  = 1  − 0.774·5-s + 0.755·7-s − 1.04·11-s + 0.277·13-s − 1.26·17-s − 0.458·19-s − 0.722·23-s − 0.400·25-s + 0.321·29-s + 1.43·31-s − 0.585·35-s + 1.15·37-s + 1.08·41-s − 0.304·43-s − 1.01·47-s − 0.428·49-s + 0.809·55-s + 1.80·59-s + 0.896·61-s − 0.214·65-s + 1.22·67-s + 1.23·71-s − 0.819·73-s − 0.789·77-s + 0.225·79-s − 1.52·83-s + 0.976·85-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

\(L(1)\) \(\approx\) \(1.323010221\)
\(L(\frac12)\) \(\approx\) \(1.323010221\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.210131723990394256438834272000, −7.72044542760917893777194766109, −6.79927285251474469747118325501, −6.10780284996259074793155577325, −5.15961748232879630934924578638, −4.46471926198148358606993603713, −3.92026424164421491252919970408, −2.73265518837680253266345736382, −2.01833638152351395471032114650, −0.60555317707886083059899131770, 0.60555317707886083059899131770, 2.01833638152351395471032114650, 2.73265518837680253266345736382, 3.92026424164421491252919970408, 4.46471926198148358606993603713, 5.15961748232879630934924578638, 6.10780284996259074793155577325, 6.79927285251474469747118325501, 7.72044542760917893777194766109, 8.210131723990394256438834272000

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