Properties

Label 2-23520-1.1-c1-0-37
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 − 3·11-s + 13-s − 15-s − 4·17-s + 19-s − 5·23-s + 25-s + 27-s + 4·29-s − 2·31-s − 3·33-s − 3·37-s + 39-s + 3·41-s + 4·43-s − 45-s + 9·47-s − 4·51-s + 3·53-s + 3·55-s + 57-s + 12·59-s − 65-s − 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s + 0.229·19-s − 1.04·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.359·31-s − 0.522·33-s − 0.493·37-s + 0.160·39-s + 0.468·41-s + 0.609·43-s − 0.149·45-s + 1.31·47-s − 0.560·51-s + 0.412·53-s + 0.404·55-s + 0.132·57-s + 1.56·59-s − 0.124·65-s − 0.488·67-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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.72866754169675, −15.29933956543530, −14.64123942793156, −14.05270338873712, −13.59109561577004, −13.13820875903714, −12.40248811398827, −12.12369406053011, −11.20399706718425, −10.88933522809182, −10.19181701988533, −9.718342096566391, −8.939642805869183, −8.468538838876533, −8.027697381510118, −7.320613206818118, −6.913444581740502, −6.047802525282225, −5.457962782757335, −4.659179857469296, −4.074061994733692, −3.496410918202341, −2.570998725179640, −2.196732298204602, −1.047169034462801, 0, 1.047169034462801, 2.196732298204602, 2.570998725179640, 3.496410918202341, 4.074061994733692, 4.659179857469296, 5.457962782757335, 6.047802525282225, 6.913444581740502, 7.320613206818118, 8.027697381510118, 8.468538838876533, 8.939642805869183, 9.718342096566391, 10.19181701988533, 10.88933522809182, 11.20399706718425, 12.12369406053011, 12.40248811398827, 13.13820875903714, 13.59109561577004, 14.05270338873712, 14.64123942793156, 15.29933956543530, 15.72866754169675

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