Properties

Degree $2$
Conductor $23520$
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 − 5-s + 9-s + 6·11-s − 4·13-s − 15-s + 2·17-s + 4·19-s + 6·23-s + 25-s + 27-s + 6·29-s + 6·33-s − 10·37-s − 4·39-s + 6·41-s + 8·43-s − 45-s + 8·47-s + 2·51-s + 6·53-s − 6·55-s + 4·57-s + 4·59-s + 4·61-s + 4·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.04·33-s − 1.64·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.809·55-s + 0.529·57-s + 0.520·59-s + 0.512·61-s + 0.496·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23520\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{23520} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.430244306\)
\(L(\frac12)\) \(\approx\) \(3.430244306\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−15.25562513929499, −14.80622736775551, −14.50498598090561, −13.84107389036670, −13.56214724040561, −12.49125245947467, −12.14105853319258, −11.94625898241940, −11.06899620915972, −10.54434877463679, −9.762207653346265, −9.334236926817027, −8.874575357374480, −8.310613464831912, −7.410580838415707, −7.177239171892598, −6.624118814970628, −5.718106112559350, −5.045214492131671, −4.316751857656293, −3.807149996495841, −3.077502240114276, −2.481322707022819, −1.388424769465531, −0.8005617773775270, 0.8005617773775270, 1.388424769465531, 2.481322707022819, 3.077502240114276, 3.807149996495841, 4.316751857656293, 5.045214492131671, 5.718106112559350, 6.624118814970628, 7.177239171892598, 7.410580838415707, 8.310613464831912, 8.874575357374480, 9.334236926817027, 9.762207653346265, 10.54434877463679, 11.06899620915972, 11.94625898241940, 12.14105853319258, 12.49125245947467, 13.56214724040561, 13.84107389036670, 14.50498598090561, 14.80622736775551, 15.25562513929499

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