Properties

Label 2-58800-1.1-c1-0-224
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 − 2·13-s + 6·17-s − 4·19-s + 4·23-s + 27-s + 6·29-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s − 8·47-s + 6·51-s − 14·53-s − 4·57-s − 4·59-s + 2·61-s + 12·67-s + 4·69-s + 12·71-s + 10·73-s − 8·79-s + 81-s + 4·83-s + 6·87-s − 6·89-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 0.840·51-s − 1.92·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.481·69-s + 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.643·87-s − 0.635·89-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.58300992868779, −14.03131972453127, −13.81750256231894, −12.94612343965135, −12.55896078373698, −12.29121455197537, −11.49873642279336, −11.06875354993230, −10.29935278229796, −10.00445082550918, −9.518520287823227, −8.866534886492247, −8.266633987334690, −8.003087742901513, −7.332563133411019, −6.611610005674423, −6.433972587897551, −5.324567573923565, −5.051185189750941, −4.408989686651921, −3.504512428111926, −3.240190318152290, −2.477962955789908, −1.750994066011071, −1.063756941581446, 0, 1.063756941581446, 1.750994066011071, 2.477962955789908, 3.240190318152290, 3.504512428111926, 4.408989686651921, 5.051185189750941, 5.324567573923565, 6.433972587897551, 6.611610005674423, 7.332563133411019, 8.003087742901513, 8.266633987334690, 8.866534886492247, 9.518520287823227, 10.00445082550918, 10.29935278229796, 11.06875354993230, 11.49873642279336, 12.29121455197537, 12.55896078373698, 12.94612343965135, 13.81750256231894, 14.03131972453127, 14.58300992868779

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