Properties

Label 2-18480-1.1-c1-0-8
Degree $2$
Conductor $18480$
Sign $1$
Analytic cond. $147.563$
Root an. cond. $12.1475$
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 + 7-s + 9-s − 11-s − 6·13-s − 15-s + 2·17-s + 21-s − 6·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 33-s − 35-s + 10·37-s − 6·39-s − 8·41-s + 8·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s + 55-s + 6·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.258·15-s + 0.485·17-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.174·33-s − 0.169·35-s + 1.64·37-s − 0.960·39-s − 1.24·41-s + 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.134·55-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18480\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(147.563\)
Root analytic conductor: \(12.1475\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{18480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.003319505\)
\(L(\frac12)\) \(\approx\) \(2.003319505\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.62357868588049, −15.09042011267171, −14.79625237315870, −14.04424626116200, −13.84818501092722, −12.96251296545875, −12.36540805939580, −12.01882004031908, −11.46912517360244, −10.64609641292438, −10.02624595454301, −9.744295131534739, −8.968657098680381, −8.185830162221285, −7.907548696924511, −7.346937286679921, −6.708663337771778, −5.869981255435066, −5.128957077130982, −4.481715618991177, −4.022732876049781, −2.952706349869976, −2.565383467676018, −1.683832338463945, −0.5638554785167849, 0.5638554785167849, 1.683832338463945, 2.565383467676018, 2.952706349869976, 4.022732876049781, 4.481715618991177, 5.128957077130982, 5.869981255435066, 6.708663337771778, 7.346937286679921, 7.907548696924511, 8.185830162221285, 8.968657098680381, 9.744295131534739, 10.02624595454301, 10.64609641292438, 11.46912517360244, 12.01882004031908, 12.36540805939580, 12.96251296545875, 13.84818501092722, 14.04424626116200, 14.79625237315870, 15.09042011267171, 15.62357868588049

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