Properties

Label 2-72e2-1.1-c1-0-42
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  + 5-s + 1.73·7-s + 1.73·11-s + 3·13-s + 4·17-s + 6.92·19-s + 8.66·23-s − 4·25-s − 29-s − 5.19·31-s + 1.73·35-s + 8·37-s + 5·41-s − 8.66·43-s − 12.1·47-s − 4·49-s + 8·53-s + 1.73·55-s + 1.73·59-s + 7·61-s + 3·65-s − 8.66·67-s − 3.46·71-s − 12·73-s + 2.99·77-s + 5.19·79-s − 8.66·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.654·7-s + 0.522·11-s + 0.832·13-s + 0.970·17-s + 1.58·19-s + 1.80·23-s − 0.800·25-s − 0.185·29-s − 0.933·31-s + 0.292·35-s + 1.31·37-s + 0.780·41-s − 1.32·43-s − 1.76·47-s − 0.571·49-s + 1.09·53-s + 0.233·55-s + 0.225·59-s + 0.896·61-s + 0.372·65-s − 1.05·67-s − 0.411·71-s − 1.40·73-s + 0.341·77-s + 0.584·79-s − 0.950·83-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

\(L(1)\) \(\approx\) \(2.924250794\)
\(L(\frac12)\) \(\approx\) \(2.924250794\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 - 8.66T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 5.19T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 8.66T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 8.66T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 3T + 97T^{2} \)
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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

−8.179886045324226786050325520905, −7.50379268684400312673491431400, −6.85460500963860429983786609745, −5.87880659801760908589403429296, −5.41298844028483740831012677680, −4.62502085878603352402825735999, −3.59377826443859072203953095189, −2.97413669370584990461976320237, −1.64950041363315760380598492516, −1.05391027046131706768994861313, 1.05391027046131706768994861313, 1.64950041363315760380598492516, 2.97413669370584990461976320237, 3.59377826443859072203953095189, 4.62502085878603352402825735999, 5.41298844028483740831012677680, 5.87880659801760908589403429296, 6.85460500963860429983786609745, 7.50379268684400312673491431400, 8.179886045324226786050325520905

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