Properties

Label 2-23520-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s + 4·11-s + 6·13-s + 15-s − 6·17-s − 4·19-s + 4·23-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·33-s + 6·37-s − 6·39-s − 6·41-s − 8·43-s − 45-s + 6·51-s + 6·53-s − 4·55-s + 4·57-s − 4·59-s − 10·61-s − 6·65-s − 8·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.960·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s + 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.744·65-s − 0.977·67-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: $-1$
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: \(1\)
Selberg data: \((2,\ 23520,\ (\ :1/2),\ -1)\)

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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.59324023975727, −15.14637131054888, −15.00387357660468, −13.90913862059487, −13.51009070256132, −13.11896818757528, −12.38210597792339, −11.78588179910542, −11.39305607264012, −10.84787329624806, −10.55388814619484, −9.549595582051079, −9.086128765271618, −8.430451967211308, −8.160573077672835, −6.997822001940043, −6.639459076233740, −6.268643179500522, −5.553597222353493, −4.518489827811720, −4.303567181337555, −3.598079847462377, −2.781358864753471, −1.675520927326735, −1.089345653411850, 0, 1.089345653411850, 1.675520927326735, 2.781358864753471, 3.598079847462377, 4.303567181337555, 4.518489827811720, 5.553597222353493, 6.268643179500522, 6.639459076233740, 6.997822001940043, 8.160573077672835, 8.430451967211308, 9.086128765271618, 9.549595582051079, 10.55388814619484, 10.84787329624806, 11.39305607264012, 11.78588179910542, 12.38210597792339, 13.11896818757528, 13.51009070256132, 13.90913862059487, 15.00387357660468, 15.14637131054888, 15.59324023975727

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