Properties

Label 2-58800-1.1-c1-0-195
Degree $2$
Conductor $58800$
Sign $-1$
Analytic cond. $469.520$
Root an. cond. $21.6684$
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 + 9-s − 4·13-s − 4·17-s + 4·19-s + 4·23-s + 27-s + 2·29-s − 8·31-s + 6·37-s − 4·39-s + 12·41-s + 4·43-s − 8·47-s − 4·51-s − 6·53-s + 4·57-s − 12·59-s + 4·61-s − 4·67-s + 4·69-s + 12·71-s − 8·73-s + 16·79-s + 81-s − 4·83-s + 2·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.10·13-s − 0.970·17-s + 0.917·19-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.640·39-s + 1.87·41-s + 0.609·43-s − 1.16·47-s − 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 0.512·61-s − 0.488·67-s + 0.481·69-s + 1.42·71-s − 0.936·73-s + 1.80·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(469.520\)
Root analytic conductor: \(21.6684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.58285395472520, −14.17119119425071, −13.60180822554929, −13.04511093499664, −12.64344732648705, −12.19723301397633, −11.48496907060772, −10.92567276403238, −10.67994884740956, −9.674437777648386, −9.402893341976816, −9.174520247869460, −8.302118046040088, −7.787251973756648, −7.365100367976231, −6.836292930095360, −6.222505602951358, −5.477233695800432, −4.902587307105388, −4.396862354911482, −3.747167601908278, −2.931850576646368, −2.581053786600828, −1.819205281853392, −0.9981022325069598, 0, 0.9981022325069598, 1.819205281853392, 2.581053786600828, 2.931850576646368, 3.747167601908278, 4.396862354911482, 4.902587307105388, 5.477233695800432, 6.222505602951358, 6.836292930095360, 7.365100367976231, 7.787251973756648, 8.302118046040088, 9.174520247869460, 9.402893341976816, 9.674437777648386, 10.67994884740956, 10.92567276403238, 11.48496907060772, 12.19723301397633, 12.64344732648705, 13.04511093499664, 13.60180822554929, 14.17119119425071, 14.58285395472520

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