Properties

Label 2-18480-1.1-c1-0-35
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 $1$

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 − 7·17-s + 5·19-s + 21-s + 23-s + 25-s − 27-s − 5·29-s + 8·31-s + 33-s − 35-s − 2·37-s + 6·39-s + 12·41-s + 11·43-s + 45-s − 8·47-s + 49-s + 7·51-s − 11·53-s − 55-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 − 1.69·17-s + 1.14·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 1.43·31-s + 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.960·39-s + 1.87·41-s + 1.67·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.980·51-s − 1.51·53-s − 0.134·55-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: \(1\)
Selberg data: \((2,\ 18480,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−16.10008841843176, −15.58104863864827, −15.01881572674131, −14.34011911219085, −13.87607949917729, −13.12903935275993, −12.79389126337932, −12.25434027108349, −11.50866116854455, −11.16380957448562, −10.41684874566037, −9.880267578463368, −9.369800869322174, −8.971362692036588, −7.886753568030484, −7.436745824163245, −6.823926003490168, −6.218325896684952, −5.617682758060527, −4.792299498135356, −4.598021504694343, −3.515713448785239, −2.558221201996797, −2.194758099003395, −0.9386306883423345, 0, 0.9386306883423345, 2.194758099003395, 2.558221201996797, 3.515713448785239, 4.598021504694343, 4.792299498135356, 5.617682758060527, 6.218325896684952, 6.823926003490168, 7.436745824163245, 7.886753568030484, 8.971362692036588, 9.369800869322174, 9.880267578463368, 10.41684874566037, 11.16380957448562, 11.50866116854455, 12.25434027108349, 12.79389126337932, 13.12903935275993, 13.87607949917729, 14.34011911219085, 15.01881572674131, 15.58104863864827, 16.10008841843176

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