Properties

Label 2-23520-1.1-c1-0-5
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 − 4·11-s + 2·13-s + 15-s + 2·17-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 2·51-s − 6·53-s + 4·55-s + 8·59-s + 2·61-s − 2·65-s − 4·67-s + 4·71-s + 10·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 1/5·25-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 + 0.937·41-s − 0.609·43-s − 0.149·45-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 1.04·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.474·71-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

\(L(1)\) \(\approx\) \(1.413335210\)
\(L(\frac12)\) \(\approx\) \(1.413335210\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.68026654536651, −15.00034293587327, −14.42026000583062, −13.70413802439138, −13.28344616870382, −12.70054836375831, −12.12178851420028, −11.73651463227078, −10.95660095589310, −10.70103031951806, −9.991905789881615, −9.594680468103642, −8.605063127315466, −8.143500841303859, −7.733726498379919, −6.933443226572415, −6.391770136096998, −5.744883660080102, −5.107494898720570, −4.571592006585672, −3.870198356932843, −3.022583509930152, −2.457080086085446, −1.291502846289273, −0.5392962175495003, 0.5392962175495003, 1.291502846289273, 2.457080086085446, 3.022583509930152, 3.870198356932843, 4.571592006585672, 5.107494898720570, 5.744883660080102, 6.391770136096998, 6.933443226572415, 7.733726498379919, 8.143500841303859, 8.605063127315466, 9.594680468103642, 9.991905789881615, 10.70103031951806, 10.95660095589310, 11.73651463227078, 12.12178851420028, 12.70054836375831, 13.28344616870382, 13.70413802439138, 14.42026000583062, 15.00034293587327, 15.68026654536651

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