Properties

Label 2-58800-1.1-c1-0-86
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 − 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: $\chi_{58800} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
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

−14.75832137650489, −13.93373614269783, −13.58102130926962, −12.95321151842072, −12.44287079554372, −12.32070253831529, −11.47304973914381, −10.85024355106505, −10.57124369612118, −10.25908870302675, −9.420952590338108, −8.981316833184409, −8.299004355987261, −7.788665358602236, −7.277761556395473, −6.708381681656203, −6.088781997749878, −5.523031108184598, −5.067241633349448, −4.397290006747318, −3.970488752189847, −2.962517722193113, −2.382035208888422, −1.871000091223573, −0.6866728649068853, 0, 0.6866728649068853, 1.871000091223573, 2.382035208888422, 2.962517722193113, 3.970488752189847, 4.397290006747318, 5.067241633349448, 5.523031108184598, 6.088781997749878, 6.708381681656203, 7.277761556395473, 7.788665358602236, 8.299004355987261, 8.981316833184409, 9.420952590338108, 10.25908870302675, 10.57124369612118, 10.85024355106505, 11.47304973914381, 12.32070253831529, 12.44287079554372, 12.95321151842072, 13.58102130926962, 13.93373614269783, 14.75832137650489

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