Properties

Degree $2$
Conductor $11760$
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 + 5-s + 9-s − 2·11-s + 2·13-s + 15-s − 4·17-s − 8·23-s + 25-s + 27-s + 2·31-s − 2·33-s + 8·37-s + 2·39-s − 2·41-s + 2·43-s + 45-s − 10·47-s − 4·51-s − 2·53-s − 2·55-s − 4·59-s − 10·61-s + 2·65-s − 2·67-s − 8·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s − 1.66·23-s + 1/5·25-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 + 0.149·45-s − 1.45·47-s − 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.244·67-s − 0.963·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11760\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{11760} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11760,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
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

−16.63651643171946, −15.85266969616492, −15.74491615221440, −14.95535621156102, −14.35283199818554, −13.81798005655849, −13.33220698497145, −12.88601355848090, −12.20492600741706, −11.44076745190849, −10.89654751034580, −10.21895002550052, −9.697568815382216, −9.139125083528583, −8.350497177958591, −8.018859057862646, −7.278125062372435, −6.346494760547037, −6.087272936631628, −5.095214170013818, −4.413288711224571, −3.726251708452576, −2.821481497836156, −2.195503052622376, −1.385261864562652, 0, 1.385261864562652, 2.195503052622376, 2.821481497836156, 3.726251708452576, 4.413288711224571, 5.095214170013818, 6.087272936631628, 6.346494760547037, 7.278125062372435, 8.018859057862646, 8.350497177958591, 9.139125083528583, 9.697568815382216, 10.21895002550052, 10.89654751034580, 11.44076745190849, 12.20492600741706, 12.88601355848090, 13.33220698497145, 13.81798005655849, 14.35283199818554, 14.95535621156102, 15.74491615221440, 15.85266969616492, 16.63651643171946

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