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Properties

Label 2-270480-1.1-c1-0-115
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 + 6·11-s + 15-s − 6·17-s − 8·19-s − 23-s + 25-s − 27-s − 6·29-s − 6·31-s − 6·33-s + 8·37-s + 10·41-s + 8·43-s − 45-s + 8·47-s + 6·51-s − 4·53-s − 6·55-s + 8·57-s − 10·59-s − 2·61-s + 4·67-s + 69-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 1.04·33-s + 1.31·37-s + 1.56·41-s + 1.21·43-s − 0.149·45-s + 1.16·47-s + 0.840·51-s − 0.549·53-s − 0.809·55-s + 1.05·57-s − 1.30·59-s − 0.256·61-s + 0.488·67-s + 0.120·69-s − 0.468·73-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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.97321524792757, −12.34836078109667, −12.27885067686408, −11.45121578744069, −11.21678665641194, −10.75909689157258, −10.61907477979156, −9.497619202195090, −9.314287001467847, −9.002785974538320, −8.441214558385605, −7.755058323020948, −7.374623190473469, −6.728569294888751, −6.450173363401226, −5.966072476167898, −5.581437662960790, −4.530553940619443, −4.332380738727605, −4.050191454224655, −3.469600395991063, −2.490184484842260, −2.075955470527405, −1.424165579225496, −0.6764296876159341, 0, 0.6764296876159341, 1.424165579225496, 2.075955470527405, 2.490184484842260, 3.469600395991063, 4.050191454224655, 4.332380738727605, 4.530553940619443, 5.581437662960790, 5.966072476167898, 6.450173363401226, 6.728569294888751, 7.374623190473469, 7.755058323020948, 8.441214558385605, 9.002785974538320, 9.314287001467847, 9.497619202195090, 10.61907477979156, 10.75909689157258, 11.21678665641194, 11.45121578744069, 12.27885067686408, 12.34836078109667, 12.97321524792757

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