Properties

Label 2-23520-1.1-c1-0-14
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 − 2·11-s + 15-s + 6·17-s − 6·19-s + 8·23-s + 25-s + 27-s + 6·29-s − 6·31-s − 2·33-s − 10·37-s − 2·41-s + 4·43-s + 45-s + 8·47-s + 6·51-s − 12·53-s − 2·55-s − 6·57-s + 12·59-s + 10·61-s + 4·67-s + 8·69-s + 2·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 1.45·17-s − 1.37·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s − 0.348·33-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 0.840·51-s − 1.64·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + 1.28·61-s + 0.488·67-s + 0.963·69-s + 0.237·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.189497388\)
\(L(\frac12)\) \(\approx\) \(3.189497388\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.42694534516031, −14.77895752839582, −14.39368637528442, −13.95562156243993, −13.23134625839817, −12.76917476462135, −12.45163051845289, −11.70633304486179, −10.76455409774901, −10.62621814804779, −9.941622637395980, −9.379655765607862, −8.698700789091885, −8.367682858149374, −7.637738748175157, −6.998378485318909, −6.558503553431835, −5.562900569227903, −5.261069273910311, −4.471502486098643, −3.643112304296137, −3.046846893944972, −2.377897391840381, −1.614834591003440, −0.7011787163990546, 0.7011787163990546, 1.614834591003440, 2.377897391840381, 3.046846893944972, 3.643112304296137, 4.471502486098643, 5.261069273910311, 5.562900569227903, 6.558503553431835, 6.998378485318909, 7.637738748175157, 8.367682858149374, 8.698700789091885, 9.379655765607862, 9.941622637395980, 10.62621814804779, 10.76455409774901, 11.70633304486179, 12.45163051845289, 12.76917476462135, 13.23134625839817, 13.95562156243993, 14.39368637528442, 14.77895752839582, 15.42694534516031

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