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Properties

Label 2-270480-1.1-c1-0-29
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s + 11-s − 4·13-s + 15-s − 6·17-s + 7·19-s + 23-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s − 2·37-s + 4·39-s − 9·41-s − 2·43-s − 45-s + 7·47-s + 6·51-s + 5·53-s − 55-s − 7·57-s + 7·59-s + 7·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s + 1.60·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 0.328·37-s + 0.640·39-s − 1.40·41-s − 0.304·43-s − 0.149·45-s + 1.02·47-s + 0.840·51-s + 0.686·53-s − 0.134·55-s − 0.927·57-s + 0.911·59-s + 0.896·61-s + 0.496·65-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: \(0\)
Selberg data: \((2,\ 270480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.246088458\)
\(L(\frac12)\) \(\approx\) \(1.246088458\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.75237124249165, −12.04712313526389, −11.89401833529245, −11.55970940418004, −10.98449212874443, −10.55947288537814, −10.01488869062170, −9.562363984434041, −9.149904229772331, −8.644851499727243, −8.050781846190604, −7.487170955297006, −7.144640810127949, −6.699664015668679, −6.269091501278020, −5.456287251316461, −5.067453132374069, −4.843046338865945, −3.874035317390294, −3.829349095510468, −2.925668079529703, −2.363541841295174, −1.789043290969675, −0.9563929464386176, −0.3686293355240894, 0.3686293355240894, 0.9563929464386176, 1.789043290969675, 2.363541841295174, 2.925668079529703, 3.829349095510468, 3.874035317390294, 4.843046338865945, 5.067453132374069, 5.456287251316461, 6.269091501278020, 6.699664015668679, 7.144640810127949, 7.487170955297006, 8.050781846190604, 8.644851499727243, 9.149904229772331, 9.562363984434041, 10.01488869062170, 10.55947288537814, 10.98449212874443, 11.55970940418004, 11.89401833529245, 12.04712313526389, 12.75237124249165

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