Properties

Label 2-58800-1.1-c1-0-110
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 + 11-s − 4·13-s − 2·17-s − 4·19-s + 3·23-s − 27-s − 29-s − 6·31-s − 33-s − 3·37-s + 4·39-s − 5·43-s + 4·47-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s + 4·61-s + 5·67-s − 3·69-s + 71-s − 10·73-s + 79-s + 81-s + 14·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.625·23-s − 0.192·27-s − 0.185·29-s − 1.07·31-s − 0.174·33-s − 0.493·37-s + 0.640·39-s − 0.762·43-s + 0.583·47-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.512·61-s + 0.610·67-s − 0.361·69-s + 0.118·71-s − 1.17·73-s + 0.112·79-s + 1/9·81-s + 1.53·83-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: Trivial
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 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.70847795205565, −14.12808542566492, −13.50241609084750, −12.90242662403635, −12.64718970366221, −12.00294713511140, −11.57963100389363, −11.05685878897384, −10.47782463214383, −10.13967058240558, −9.435899656592275, −8.924066248713585, −8.530981008427332, −7.535423618790010, −7.376921494384460, −6.600263843487893, −6.290354027846746, −5.440684926987634, −5.048639010925029, −4.454927991543419, −3.847769403888180, −3.156345235643225, −2.250353116554415, −1.843197928847298, −0.7765305004596191, 0, 0.7765305004596191, 1.843197928847298, 2.250353116554415, 3.156345235643225, 3.847769403888180, 4.454927991543419, 5.048639010925029, 5.440684926987634, 6.290354027846746, 6.600263843487893, 7.376921494384460, 7.535423618790010, 8.530981008427332, 8.924066248713585, 9.435899656592275, 10.13967058240558, 10.47782463214383, 11.05685878897384, 11.57963100389363, 12.00294713511140, 12.64718970366221, 12.90242662403635, 13.50241609084750, 14.12808542566492, 14.70847795205565

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