Properties

Label 2-23520-1.1-c1-0-16
Degree $2$
Conductor $23520$
Sign $1$
Analytic cond. $187.808$
Root an. cond. $13.7043$
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 − 5-s + 9-s + 2·13-s − 15-s + 6·17-s − 4·23-s + 25-s + 27-s − 2·29-s + 8·31-s + 6·37-s + 2·39-s + 6·41-s + 12·43-s − 45-s + 12·47-s + 6·51-s − 10·53-s − 8·59-s + 10·61-s − 2·65-s − 12·67-s − 4·69-s + 8·71-s − 10·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s − 0.149·45-s + 1.75·47-s + 0.840·51-s − 1.37·53-s − 1.04·59-s + 1.28·61-s − 0.248·65-s − 1.46·67-s − 0.481·69-s + 0.949·71-s − 1.17·73-s + 0.115·75-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: $1$
Analytic conductor: \(187.808\)
Root analytic conductor: \(13.7043\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{23520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.083577649\)
\(L(\frac12)\) \(\approx\) \(3.083577649\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.62129339731175, −14.84045488154229, −14.29717971951819, −13.99196150774800, −13.37381636769435, −12.67392914867343, −12.23438529770749, −11.79092003903630, −10.96162095284614, −10.61179030190351, −9.805624170201206, −9.445346751108246, −8.758093425305613, −8.083961291717067, −7.723226499255893, −7.268934919078152, −6.236874394556440, −5.929598584070976, −5.092188268674299, −4.139202353129676, −3.965668006805517, −2.981532555706256, −2.542624542873271, −1.424031327167036, −0.7373857832357376, 0.7373857832357376, 1.424031327167036, 2.542624542873271, 2.981532555706256, 3.965668006805517, 4.139202353129676, 5.092188268674299, 5.929598584070976, 6.236874394556440, 7.268934919078152, 7.723226499255893, 8.083961291717067, 8.758093425305613, 9.445346751108246, 9.805624170201206, 10.61179030190351, 10.96162095284614, 11.79092003903630, 12.23438529770749, 12.67392914867343, 13.37381636769435, 13.99196150774800, 14.29717971951819, 14.84045488154229, 15.62129339731175

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