Properties

Label 2-58800-1.1-c1-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s − 6·13-s + 6·17-s − 2·19-s + 6·23-s − 27-s + 6·29-s + 2·31-s + 4·33-s − 2·37-s + 6·39-s + 2·41-s − 8·43-s − 6·51-s − 4·53-s + 2·57-s − 8·59-s + 8·61-s − 8·67-s − 6·69-s − 12·71-s + 6·73-s + 8·79-s + 81-s − 4·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 0.312·41-s − 1.21·43-s − 0.840·51-s − 0.549·53-s + 0.264·57-s − 1.04·59-s + 1.02·61-s − 0.977·67-s − 0.722·69-s − 1.42·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-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: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.023673070\)
\(L(\frac12)\) \(\approx\) \(1.023673070\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.36335667142181, −13.85675412453843, −13.16054685034241, −12.77972772196014, −12.20583113187553, −12.00269138199216, −11.29270618276250, −10.63367327154185, −10.28956317273830, −9.862399483407987, −9.361890856991963, −8.546807088478562, −7.999665870452727, −7.528715470453289, −7.031424798066736, −6.483367290492841, −5.695604620091589, −5.222194463976186, −4.833936014018039, −4.316310739100832, −3.161165941855147, −2.922753873882461, −2.119568106694343, −1.231710446029194, −0.3786693438550003, 0.3786693438550003, 1.231710446029194, 2.119568106694343, 2.922753873882461, 3.161165941855147, 4.316310739100832, 4.833936014018039, 5.222194463976186, 5.695604620091589, 6.483367290492841, 7.031424798066736, 7.528715470453289, 7.999665870452727, 8.546807088478562, 9.361890856991963, 9.862399483407987, 10.28956317273830, 10.63367327154185, 11.29270618276250, 12.00269138199216, 12.20583113187553, 12.77972772196014, 13.16054685034241, 13.85675412453843, 14.36335667142181

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