Properties

Label 2-54720-1.1-c1-0-62
Degree $2$
Conductor $54720$
Sign $1$
Analytic cond. $436.941$
Root an. cond. $20.9031$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·7-s − 4·11-s + 2·13-s + 2·17-s + 19-s + 8·23-s + 25-s + 6·29-s + 4·31-s + 4·35-s + 10·37-s + 2·41-s − 12·43-s + 9·49-s + 6·53-s − 4·55-s + 10·61-s + 2·65-s + 4·67-s + 8·71-s + 2·73-s − 16·77-s − 12·79-s − 8·83-s + 2·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.676·35-s + 1.64·37-s + 0.312·41-s − 1.82·43-s + 9/7·49-s + 0.824·53-s − 0.539·55-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s + 0.234·73-s − 1.82·77-s − 1.35·79-s − 0.878·83-s + 0.216·85-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54720\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(436.941\)
Root analytic conductor: \(20.9031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.199762894\)
\(L(\frac12)\) \(\approx\) \(4.199762894\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.51473399612373, −13.92756184945375, −13.30573360604197, −13.09771279969969, −12.43724226787594, −11.67124980001757, −11.34788081530058, −10.92440042559187, −10.21073162368345, −10.02501011962990, −9.152717110184909, −8.581218964624872, −8.096313419083366, −7.837938130968466, −7.008068556040063, −6.569291064592833, −5.613965271452468, −5.371755919710667, −4.749797052447261, −4.349996094134027, −3.307798651589949, −2.736255385752649, −2.139611158292116, −1.247854481319545, −0.8065432770684472, 0.8065432770684472, 1.247854481319545, 2.139611158292116, 2.736255385752649, 3.307798651589949, 4.349996094134027, 4.749797052447261, 5.371755919710667, 5.613965271452468, 6.569291064592833, 7.008068556040063, 7.837938130968466, 8.096313419083366, 8.581218964624872, 9.152717110184909, 10.02501011962990, 10.21073162368345, 10.92440042559187, 11.34788081530058, 11.67124980001757, 12.43724226787594, 13.09771279969969, 13.30573360604197, 13.92756184945375, 14.51473399612373

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