Properties

Label 2-47040-1.1-c1-0-168
Degree $2$
Conductor $47040$
Sign $-1$
Analytic cond. $375.616$
Root an. cond. $19.3808$
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 + 2·13-s − 15-s + 6·17-s − 4·19-s + 25-s + 27-s + 6·29-s + 4·31-s − 2·37-s + 2·39-s − 6·41-s − 8·43-s − 45-s + 12·47-s + 6·51-s − 6·53-s − 4·57-s − 12·59-s + 2·61-s − 2·65-s − 8·67-s − 14·73-s + 75-s − 16·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.75·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.63·73-s + 0.115·75-s − 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47040\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(375.616\)
Root analytic conductor: \(19.3808\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.04436674794512, −14.27208055695945, −13.85867760956367, −13.44052451244976, −12.77157133368668, −12.14774067095322, −12.01676513809551, −11.21701047717642, −10.58202602047985, −10.18436737873084, −9.726503758934634, −8.830915574212316, −8.609531122680396, −8.021255042064888, −7.548898318384794, −6.913226267826272, −6.305875378907583, −5.758351633271600, −4.956233909011682, −4.398766292466767, −3.795550673474404, −3.123175777548146, −2.724953823588603, −1.659367268736002, −1.123560417712240, 0, 1.123560417712240, 1.659367268736002, 2.724953823588603, 3.123175777548146, 3.795550673474404, 4.398766292466767, 4.956233909011682, 5.758351633271600, 6.305875378907583, 6.913226267826272, 7.548898318384794, 8.021255042064888, 8.609531122680396, 8.830915574212316, 9.726503758934634, 10.18436737873084, 10.58202602047985, 11.21701047717642, 12.01676513809551, 12.14774067095322, 12.77157133368668, 13.44052451244976, 13.85867760956367, 14.27208055695945, 15.04436674794512

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