Properties

Label 2-58800-1.1-c1-0-30
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 − 6·13-s + 2·17-s + 4·19-s − 4·23-s + 27-s − 5·31-s − 2·33-s + 7·37-s − 6·39-s − 10·41-s + 9·43-s − 6·47-s + 2·51-s + 14·53-s + 4·57-s − 10·59-s − 15·61-s + 4·67-s − 4·69-s − 2·71-s − 9·73-s + 79-s + 81-s − 10·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 0.192·27-s − 0.898·31-s − 0.348·33-s + 1.15·37-s − 0.960·39-s − 1.56·41-s + 1.37·43-s − 0.875·47-s + 0.280·51-s + 1.92·53-s + 0.529·57-s − 1.30·59-s − 1.92·61-s + 0.488·67-s − 0.481·69-s − 0.237·71-s − 1.05·73-s + 0.112·79-s + 1/9·81-s − 1.09·83-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

\(L(1)\) \(\approx\) \(1.743365566\)
\(L(\frac12)\) \(\approx\) \(1.743365566\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−14.32478225008524, −13.89494614951459, −13.37204129502411, −12.85996113021338, −12.19693532742077, −12.03032306951447, −11.34956610544942, −10.57693349599411, −10.20626102883961, −9.616186329484169, −9.363315911940974, −8.637501796509140, −7.952066342212043, −7.563626646950884, −7.266448716218981, −6.520448744435571, −5.666533402157480, −5.352771038713517, −4.599535006074512, −4.140863688423843, −3.222228685157932, −2.850446574832318, −2.162130917390848, −1.497241308501971, −0.4198022588668192, 0.4198022588668192, 1.497241308501971, 2.162130917390848, 2.850446574832318, 3.222228685157932, 4.140863688423843, 4.599535006074512, 5.352771038713517, 5.666533402157480, 6.520448744435571, 7.266448716218981, 7.563626646950884, 7.952066342212043, 8.637501796509140, 9.363315911940974, 9.616186329484169, 10.20626102883961, 10.57693349599411, 11.34956610544942, 12.03032306951447, 12.19693532742077, 12.85996113021338, 13.37204129502411, 13.89494614951459, 14.32478225008524

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