Properties

Label 2-58800-1.1-c1-0-4
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 − 5·11-s + 4·17-s − 8·19-s − 4·23-s − 27-s − 5·29-s − 3·31-s + 5·33-s + 4·37-s + 2·43-s − 6·47-s − 4·51-s + 9·53-s + 8·57-s + 11·59-s − 6·61-s − 2·67-s + 4·69-s − 2·71-s − 10·73-s − 3·79-s + 81-s − 7·83-s + 5·87-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.50·11-s + 0.970·17-s − 1.83·19-s − 0.834·23-s − 0.192·27-s − 0.928·29-s − 0.538·31-s + 0.870·33-s + 0.657·37-s + 0.304·43-s − 0.875·47-s − 0.560·51-s + 1.23·53-s + 1.05·57-s + 1.43·59-s − 0.768·61-s − 0.244·67-s + 0.481·69-s − 0.237·71-s − 1.17·73-s − 0.337·79-s + 1/9·81-s − 0.768·83-s + 0.536·87-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: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3739072115\)
\(L(\frac12)\) \(\approx\) \(0.3739072115\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.57240761155172, −13.66342962740502, −13.18868901740414, −12.82459241456833, −12.41632128821051, −11.78766722230618, −11.21266471798854, −10.77588294453110, −10.20083228860551, −10.00425541414318, −9.244443409587305, −8.431765151171698, −8.211594089250758, −7.395483743580642, −7.192924245636370, −6.186119024476900, −5.895982267338470, −5.355061909836361, −4.743782217284167, −4.115862797634674, −3.543273948830882, −2.622130106939343, −2.158596880601819, −1.321367080470983, −0.2166290860246673, 0.2166290860246673, 1.321367080470983, 2.158596880601819, 2.622130106939343, 3.543273948830882, 4.115862797634674, 4.743782217284167, 5.355061909836361, 5.895982267338470, 6.186119024476900, 7.192924245636370, 7.395483743580642, 8.211594089250758, 8.431765151171698, 9.244443409587305, 10.00425541414318, 10.20083228860551, 10.77588294453110, 11.21266471798854, 11.78766722230618, 12.41632128821051, 12.82459241456833, 13.18868901740414, 13.66342962740502, 14.57240761155172

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