Properties

Label 2-58800-1.1-c1-0-101
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 − 3·11-s − 4·13-s − 4·19-s − 27-s + 9·29-s − 31-s + 3·33-s − 8·37-s + 4·39-s − 10·43-s + 6·47-s + 3·53-s + 4·57-s + 3·59-s + 10·61-s − 10·67-s + 6·71-s + 2·73-s + 79-s + 81-s + 9·83-s − 9·87-s − 6·89-s + 93-s − 97-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.917·19-s − 0.192·27-s + 1.67·29-s − 0.179·31-s + 0.522·33-s − 1.31·37-s + 0.640·39-s − 1.52·43-s + 0.875·47-s + 0.412·53-s + 0.529·57-s + 0.390·59-s + 1.28·61-s − 1.22·67-s + 0.712·71-s + 0.234·73-s + 0.112·79-s + 1/9·81-s + 0.987·83-s − 0.964·87-s − 0.635·89-s + 0.103·93-s − 0.101·97-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.58950171278508, −14.10739884966583, −13.47424691733562, −13.05773700891463, −12.42308447509419, −12.14237396565438, −11.63800908324853, −10.95053544315158, −10.40880737491539, −10.15723941841733, −9.659279268185910, −8.726432865098171, −8.498384628402666, −7.734666433095428, −7.252483377927369, −6.657657183278970, −6.251818197774357, −5.338149840833949, −5.112708331389608, −4.529229095602766, −3.842021948431386, −3.026128579152339, −2.398697731591226, −1.804946701890767, −0.7357490882358846, 0, 0.7357490882358846, 1.804946701890767, 2.398697731591226, 3.026128579152339, 3.842021948431386, 4.529229095602766, 5.112708331389608, 5.338149840833949, 6.251818197774357, 6.657657183278970, 7.252483377927369, 7.734666433095428, 8.498384628402666, 8.726432865098171, 9.659279268185910, 10.15723941841733, 10.40880737491539, 10.95053544315158, 11.63800908324853, 12.14237396565438, 12.42308447509419, 13.05773700891463, 13.47424691733562, 14.10739884966583, 14.58950171278508

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