Properties

Label 2-270480-1.1-c1-0-236
Degree $2$
Conductor $270480$
Sign $-1$
Analytic cond. $2159.79$
Root an. cond. $46.4735$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 4·13-s + 15-s + 2·19-s − 23-s + 25-s + 27-s + 6·29-s + 2·31-s − 10·37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 6·47-s − 6·53-s + 2·57-s + 12·59-s + 10·61-s + 4·65-s + 4·67-s − 69-s − 14·73-s + 75-s − 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.10·13-s + 0.258·15-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 0.488·67-s − 0.120·69-s − 1.63·73-s + 0.115·75-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270480\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2159.79\)
Root analytic conductor: \(46.4735\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 270480,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.08146761615733, −12.63199263093202, −12.11044736263909, −11.64886915499620, −11.19701147583091, −10.52074309315884, −10.27033060402042, −9.824745245296897, −9.261214361265809, −8.717231398823558, −8.461691598725053, −8.069168151939458, −7.284842886791161, −6.952995754744127, −6.441607049268174, −5.898629270942905, −5.411562865519125, −4.882452853160453, −4.279638981522740, −3.643346475323304, −3.360597115054769, −2.587016489630880, −2.207119327188814, −1.323396923946691, −1.095827890429017, 0, 1.095827890429017, 1.323396923946691, 2.207119327188814, 2.587016489630880, 3.360597115054769, 3.643346475323304, 4.279638981522740, 4.882452853160453, 5.411562865519125, 5.898629270942905, 6.441607049268174, 6.952995754744127, 7.284842886791161, 8.069168151939458, 8.461691598725053, 8.717231398823558, 9.261214361265809, 9.824745245296897, 10.27033060402042, 10.52074309315884, 11.19701147583091, 11.64886915499620, 12.11044736263909, 12.63199263093202, 13.08146761615733

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