Properties

Degree $2$
Conductor $58800$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·11-s − 2·13-s + 2·17-s − 4·19-s − 8·23-s + 27-s + 6·29-s − 8·31-s − 4·33-s + 2·37-s − 2·39-s − 2·41-s − 12·43-s + 8·47-s + 2·51-s − 6·53-s − 4·57-s + 4·59-s + 2·61-s + 12·67-s − 8·69-s − 8·71-s − 14·73-s + 81-s − 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.82·43-s + 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.963·69-s − 0.949·71-s − 1.63·73-s + 1/9·81-s − 1.31·83-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: $1$
Motivic weight: \(1\)
Character: $\chi_{58800} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 58800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.134117521\)
\(L(\frac12)\) \(\approx\) \(1.134117521\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.39512322072031, −13.85050966283458, −13.33886878779463, −12.81515897526495, −12.44933792163522, −11.87380388796275, −11.30821677638502, −10.53811978113710, −10.21441473300928, −9.859835645958285, −9.173249029371746, −8.471299946825011, −8.158596899271208, −7.686522192087608, −7.064821832737582, −6.532614355215204, −5.718840745855096, −5.360441807003745, −4.564957725509854, −4.107302236288485, −3.347777889926922, −2.741589976136632, −2.149674366904456, −1.566009537078945, −0.3291138295401244, 0.3291138295401244, 1.566009537078945, 2.149674366904456, 2.741589976136632, 3.347777889926922, 4.107302236288485, 4.564957725509854, 5.360441807003745, 5.718840745855096, 6.532614355215204, 7.064821832737582, 7.686522192087608, 8.158596899271208, 8.471299946825011, 9.173249029371746, 9.859835645958285, 10.21441473300928, 10.53811978113710, 11.30821677638502, 11.87380388796275, 12.44933792163522, 12.81515897526495, 13.33886878779463, 13.85050966283458, 14.39512322072031

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