Properties

Label 2-58800-1.1-c1-0-67
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 − 2·11-s − 4·13-s + 2·17-s + 2·23-s + 27-s + 6·29-s + 4·31-s − 2·33-s + 10·37-s − 4·39-s + 2·41-s + 4·43-s + 8·47-s + 2·51-s − 10·53-s + 4·59-s − 8·61-s + 8·67-s + 2·69-s − 6·71-s + 4·73-s − 4·79-s + 81-s − 4·83-s + 6·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.417·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.348·33-s + 1.64·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 0.280·51-s − 1.37·53-s + 0.520·59-s − 1.02·61-s + 0.977·67-s + 0.240·69-s − 0.712·71-s + 0.468·73-s − 0.450·79-s + 1/9·81-s − 0.439·83-s + 0.643·87-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

\(L(1)\) \(\approx\) \(2.865535383\)
\(L(\frac12)\) \(\approx\) \(2.865535383\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−14.30085853291275, −13.96643510263319, −13.29351818932954, −12.87040986644984, −12.30263737341042, −11.99775012472145, −11.22017503043803, −10.70982991653176, −10.16612855442228, −9.643272498419478, −9.336240646548879, −8.541295030496020, −8.113572373708734, −7.527407735651244, −7.240391486348034, −6.407700898407460, −5.923089873841759, −5.130778431030683, −4.665049427348490, −4.147859748301046, −3.275301587847227, −2.650264471162388, −2.395266230781973, −1.321637736179472, −0.5814864070986525, 0.5814864070986525, 1.321637736179472, 2.395266230781973, 2.650264471162388, 3.275301587847227, 4.147859748301046, 4.665049427348490, 5.130778431030683, 5.923089873841759, 6.407700898407460, 7.240391486348034, 7.527407735651244, 8.113572373708734, 8.541295030496020, 9.336240646548879, 9.643272498419478, 10.16612855442228, 10.70982991653176, 11.22017503043803, 11.99775012472145, 12.30263737341042, 12.87040986644984, 13.29351818932954, 13.96643510263319, 14.30085853291275

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