Properties

Label 2-72e2-1.1-c1-0-29
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 − 4.37·7-s + 3.58·11-s + 6.58·13-s − 1.73·17-s − 2.55·19-s + 7.58·23-s + 2.00·25-s + 6.10·29-s − 8.75·31-s − 11.5·35-s − 2.58·37-s + 1.82·41-s + 2.55·43-s + 8·47-s + 12.1·49-s + 1.82·53-s + 9.47·55-s − 8·59-s + 1.41·61-s + 17.4·65-s + 2.55·67-s + 0.417·71-s − 6.16·73-s − 15.6·77-s − 9.47·79-s − 15.1·83-s + ⋯
L(s)  = 1  + 1.18·5-s − 1.65·7-s + 1.08·11-s + 1.82·13-s − 0.420·17-s − 0.585·19-s + 1.58·23-s + 0.400·25-s + 1.13·29-s − 1.57·31-s − 1.95·35-s − 0.424·37-s + 0.285·41-s + 0.388·43-s + 1.16·47-s + 1.73·49-s + 0.251·53-s + 1.27·55-s − 1.04·59-s + 0.181·61-s + 2.16·65-s + 0.311·67-s + 0.0495·71-s − 0.721·73-s − 1.78·77-s − 1.06·79-s − 1.66·83-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

\(L(1)\) \(\approx\) \(2.402008306\)
\(L(\frac12)\) \(\approx\) \(2.402008306\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.37T + 7T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
13 \( 1 - 6.58T + 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 - 6.10T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 - 0.417T + 71T^{2} \)
73 \( 1 + 6.16T + 73T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 5.16T + 97T^{2} \)
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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

−8.592895372863227820007172999931, −7.12362760900270591994553885741, −6.67718098003480573951386932004, −6.02790714898363950758915254175, −5.72011143695307138049901355607, −4.40103592724966271160538125754, −3.58848269147888325155310312929, −2.95422384906447513315934810649, −1.83843752920138279795943683168, −0.873155565318438683225002008907, 0.873155565318438683225002008907, 1.83843752920138279795943683168, 2.95422384906447513315934810649, 3.58848269147888325155310312929, 4.40103592724966271160538125754, 5.72011143695307138049901355607, 6.02790714898363950758915254175, 6.67718098003480573951386932004, 7.12362760900270591994553885741, 8.592895372863227820007172999931

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