Properties

Degree $2$
Conductor $58800$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·11-s − 2·13-s + 4·17-s + 8·23-s − 27-s + 2·31-s + 2·33-s − 8·37-s + 2·39-s − 2·41-s − 2·43-s + 10·47-s − 4·51-s + 2·53-s − 4·59-s − 10·61-s + 2·67-s − 8·69-s + 12·71-s − 10·73-s − 16·79-s + 81-s + 16·83-s + 14·89-s − 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s + 1.66·23-s − 0.192·27-s + 0.359·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s − 0.312·41-s − 0.304·43-s + 1.45·47-s − 0.560·51-s + 0.274·53-s − 0.520·59-s − 1.28·61-s + 0.244·67-s − 0.963·69-s + 1.42·71-s − 1.17·73-s − 1.80·79-s + 1/9·81-s + 1.75·83-s + 1.48·89-s − 0.207·93-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{58800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.67193731325115, −14.00313469928911, −13.57664754420637, −13.00370239432988, −12.46898049493291, −12.09949500031759, −11.64231294208160, −10.87641296744209, −10.58968047641242, −10.11373552866121, −9.483195807162827, −8.977358265092181, −8.374313049178115, −7.666146769569592, −7.298366521545048, −6.754573381142410, −6.118523031176463, −5.370376761131070, −5.158131423219626, −4.539555614333121, −3.740292549243682, −3.073282918140951, −2.524857016771351, −1.586271667304367, −0.8950729294548966, 0, 0.8950729294548966, 1.586271667304367, 2.524857016771351, 3.073282918140951, 3.740292549243682, 4.539555614333121, 5.158131423219626, 5.370376761131070, 6.118523031176463, 6.754573381142410, 7.298366521545048, 7.666146769569592, 8.374313049178115, 8.977358265092181, 9.483195807162827, 10.11373552866121, 10.58968047641242, 10.87641296744209, 11.64231294208160, 12.09949500031759, 12.46898049493291, 13.00370239432988, 13.57664754420637, 14.00313469928911, 14.67193731325115

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