Properties

Label 2-52800-1.1-c1-0-63
Degree $2$
Conductor $52800$
Sign $-1$
Analytic cond. $421.610$
Root an. cond. $20.5331$
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 − 4·7-s + 9-s − 11-s − 2·13-s + 2·17-s + 4·21-s − 8·23-s − 27-s + 6·29-s − 8·31-s + 33-s + 6·37-s + 2·39-s − 2·41-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 4·59-s − 6·61-s − 4·63-s − 4·67-s + 8·69-s + 14·73-s + 4·77-s − 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s − 0.312·41-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.503·63-s − 0.488·67-s + 0.963·69-s + 1.63·73-s + 0.455·77-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52800\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(421.610\)
Root analytic conductor: \(20.5331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 52800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.73572378873991, −14.19609416530022, −13.53905331459395, −13.15457494416960, −12.60969112414061, −12.11994723550853, −11.91315722457256, −11.04078955811806, −10.54509515567021, −10.01452192893950, −9.634370668082810, −9.283585528624298, −8.321040311416481, −7.933156653708424, −7.161460094699055, −6.775512363133880, −6.123800990136777, −5.772655780519693, −5.130799897107314, −4.402195027140824, −3.770888063849795, −3.198404523096050, −2.512134435758316, −1.771334806841168, −0.6750611742928177, 0, 0.6750611742928177, 1.771334806841168, 2.512134435758316, 3.198404523096050, 3.770888063849795, 4.402195027140824, 5.130799897107314, 5.772655780519693, 6.123800990136777, 6.775512363133880, 7.161460094699055, 7.933156653708424, 8.321040311416481, 9.283585528624298, 9.634370668082810, 10.01452192893950, 10.54509515567021, 11.04078955811806, 11.91315722457256, 12.11994723550853, 12.60969112414061, 13.15457494416960, 13.53905331459395, 14.19609416530022, 14.73572378873991

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