Properties

Label 2-58800-1.1-c1-0-35
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 − 5·11-s − 2·13-s − 2·17-s + 6·19-s + 23-s − 27-s + 3·29-s + 4·31-s + 5·33-s + 5·37-s + 2·39-s − 4·41-s − 7·43-s + 10·47-s + 2·51-s − 2·53-s − 6·57-s + 10·59-s − 8·61-s + 7·67-s − 69-s + 3·71-s + 2·73-s + 11·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.485·17-s + 1.37·19-s + 0.208·23-s − 0.192·27-s + 0.557·29-s + 0.718·31-s + 0.870·33-s + 0.821·37-s + 0.320·39-s − 0.624·41-s − 1.06·43-s + 1.45·47-s + 0.280·51-s − 0.274·53-s − 0.794·57-s + 1.30·59-s − 1.02·61-s + 0.855·67-s − 0.120·69-s + 0.356·71-s + 0.234·73-s + 1.23·79-s + 1/9·81-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: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.281102493\)
\(L(\frac12)\) \(\approx\) \(1.281102493\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.10048907317906, −13.90692240981662, −13.15073314288777, −12.95483963677702, −12.20257519380855, −11.83794386610285, −11.32503070385015, −10.68553602560419, −10.36154672822061, −9.703779720467435, −9.444952117155521, −8.463550517847605, −8.124795629651047, −7.473353805332906, −7.041560497712162, −6.458380641915411, −5.697566980463875, −5.271172072597449, −4.835640033393848, −4.235395840533627, −3.337545162287687, −2.734024144910651, −2.201492763548907, −1.180805957155866, −0.4359024093387610, 0.4359024093387610, 1.180805957155866, 2.201492763548907, 2.734024144910651, 3.337545162287687, 4.235395840533627, 4.835640033393848, 5.271172072597449, 5.697566980463875, 6.458380641915411, 7.041560497712162, 7.473353805332906, 8.124795629651047, 8.463550517847605, 9.444952117155521, 9.703779720467435, 10.36154672822061, 10.68553602560419, 11.32503070385015, 11.83794386610285, 12.20257519380855, 12.95483963677702, 13.15073314288777, 13.90692240981662, 14.10048907317906

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