Properties

Label 2-58800-1.1-c1-0-19
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 − 3·11-s + 4·13-s − 4·19-s + 8·23-s − 27-s − 3·29-s − 5·31-s + 3·33-s − 8·37-s − 4·39-s − 8·41-s + 6·43-s − 10·47-s − 9·53-s + 4·57-s − 5·59-s + 10·61-s + 6·67-s − 8·69-s − 10·71-s + 2·73-s − 11·79-s + 81-s − 7·83-s + 3·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.917·19-s + 1.66·23-s − 0.192·27-s − 0.557·29-s − 0.898·31-s + 0.522·33-s − 1.31·37-s − 0.640·39-s − 1.24·41-s + 0.914·43-s − 1.45·47-s − 1.23·53-s + 0.529·57-s − 0.650·59-s + 1.28·61-s + 0.733·67-s − 0.963·69-s − 1.18·71-s + 0.234·73-s − 1.23·79-s + 1/9·81-s − 0.768·83-s + 0.321·87-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

\(L(1)\) \(\approx\) \(0.9690771223\)
\(L(\frac12)\) \(\approx\) \(0.9690771223\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.45613879937333, −13.61954311550585, −13.24912556787692, −12.82159646765472, −12.48548679862112, −11.66599352906850, −11.16012849882923, −10.84911056071702, −10.43054792632505, −9.805223689003891, −9.102242640330812, −8.678230634373135, −8.132180505851709, −7.525350829269028, −6.797432250884835, −6.583806174006519, −5.720057180322384, −5.339818200863780, −4.809995669043709, −4.101196327175359, −3.412237638906809, −2.899526563136672, −1.901647124216141, −1.390468943213349, −0.3520502707267678, 0.3520502707267678, 1.390468943213349, 1.901647124216141, 2.899526563136672, 3.412237638906809, 4.101196327175359, 4.809995669043709, 5.339818200863780, 5.720057180322384, 6.583806174006519, 6.797432250884835, 7.525350829269028, 8.132180505851709, 8.678230634373135, 9.102242640330812, 9.805223689003891, 10.43054792632505, 10.84911056071702, 11.16012849882923, 11.66599352906850, 12.48548679862112, 12.82159646765472, 13.24912556787692, 13.61954311550585, 14.45613879937333

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