Properties

Label 2-58800-1.1-c1-0-39
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 − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 27-s − 2·29-s + 8·31-s + 4·33-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s − 2·51-s + 10·53-s − 4·57-s − 12·59-s − 6·61-s + 12·67-s − 6·73-s + 8·79-s + 81-s − 4·83-s + 2·87-s − 2·89-s − 8·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.768·61-s + 1.46·67-s − 0.702·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s − 0.211·89-s − 0.829·93-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

\(L(1)\) \(\approx\) \(1.439422377\)
\(L(\frac12)\) \(\approx\) \(1.439422377\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.24606905751142, −13.70522007723011, −13.42386733444841, −12.69112420401850, −12.27075771823837, −11.90746284374274, −11.23790778218500, −10.80142666715803, −10.22082975979885, −9.821417426940685, −9.373830809401450, −8.559938161386276, −7.975826797662010, −7.560615517965568, −7.062674648522052, −6.390442399044493, −5.687269963651377, −5.383068535834802, −4.742769202976670, −4.251147338656392, −3.323927891592937, −2.801708559383688, −2.136471367556430, −1.190552326090878, −0.4634921744827620, 0.4634921744827620, 1.190552326090878, 2.136471367556430, 2.801708559383688, 3.323927891592937, 4.251147338656392, 4.742769202976670, 5.383068535834802, 5.687269963651377, 6.390442399044493, 7.062674648522052, 7.560615517965568, 7.975826797662010, 8.559938161386276, 9.373830809401450, 9.821417426940685, 10.22082975979885, 10.80142666715803, 11.23790778218500, 11.90746284374274, 12.27075771823837, 12.69112420401850, 13.42386733444841, 13.70522007723011, 14.24606905751142

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