Properties

Degree $2$
Conductor $58800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·11-s + 2·13-s − 4·17-s + 8·23-s + 27-s − 2·31-s − 2·33-s − 8·37-s + 2·39-s + 2·41-s − 2·43-s − 10·47-s − 4·51-s + 2·53-s + 4·59-s + 10·61-s + 2·67-s + 8·69-s + 12·71-s + 10·73-s − 16·79-s + 81-s − 16·83-s − 14·89-s − 2·93-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.970·17-s + 1.66·23-s + 0.192·27-s − 0.359·31-s − 0.348·33-s − 1.31·37-s + 0.320·39-s + 0.312·41-s − 0.304·43-s − 1.45·47-s − 0.560·51-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.244·67-s + 0.963·69-s + 1.42·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s − 1.75·83-s − 1.48·89-s − 0.207·93-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: $1$
Motivic weight: \(1\)
Character: $\chi_{58800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.587254788\)
\(L(\frac12)\) \(\approx\) \(2.587254788\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.19817193892655, −13.95063655649613, −13.17581254040843, −12.90947419531165, −12.63873116997995, −11.63521907024192, −11.20198287596510, −10.91642590742962, −10.02214142364829, −9.900250566666797, −8.911680618812477, −8.728685755117431, −8.288169759633086, −7.491508221035230, −7.028705215549728, −6.597530379944704, −5.855634191377192, −5.091147218897041, −4.813776581871488, −3.922980662755433, −3.425052566832367, −2.789387547524379, −2.144942332165920, −1.448106238590092, −0.5292822448578361, 0.5292822448578361, 1.448106238590092, 2.144942332165920, 2.789387547524379, 3.425052566832367, 3.922980662755433, 4.813776581871488, 5.091147218897041, 5.855634191377192, 6.597530379944704, 7.028705215549728, 7.491508221035230, 8.288169759633086, 8.728685755117431, 8.911680618812477, 9.900250566666797, 10.02214142364829, 10.91642590742962, 11.20198287596510, 11.63521907024192, 12.63873116997995, 12.90947419531165, 13.17581254040843, 13.95063655649613, 14.19817193892655

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