Properties

Label 2-23520-1.1-c1-0-39
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 − 2·13-s − 15-s − 6·17-s + 4·19-s + 8·23-s + 25-s + 27-s − 2·29-s − 4·31-s + 10·37-s − 2·39-s − 2·41-s − 4·43-s − 45-s − 8·47-s − 6·51-s − 2·53-s + 4·57-s − 8·59-s + 2·61-s + 2·65-s − 12·67-s + 8·69-s + 8·71-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 + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 1.64·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.529·57-s − 1.04·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.963·69-s + 0.949·71-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: $\chi_{23520} (1, \cdot )$
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.67076595921886, −14.99087358905120, −14.85044013693951, −14.19475998506890, −13.30675089584359, −13.25912145608141, −12.58107198578415, −11.90502267055932, −11.22141025749023, −11.02276151365479, −10.19485280933149, −9.418578850224584, −9.206096157947180, −8.552161748005874, −7.829543383169509, −7.425337583213587, −6.789587633714783, −6.268542152204449, −5.191478796219789, −4.820563754643003, −4.114339527627141, −3.332715800364478, −2.801955573497859, −2.024053117413185, −1.103165423145101, 0, 1.103165423145101, 2.024053117413185, 2.801955573497859, 3.332715800364478, 4.114339527627141, 4.820563754643003, 5.191478796219789, 6.268542152204449, 6.789587633714783, 7.425337583213587, 7.829543383169509, 8.552161748005874, 9.206096157947180, 9.418578850224584, 10.19485280933149, 11.02276151365479, 11.22141025749023, 11.90502267055932, 12.58107198578415, 13.25912145608141, 13.30675089584359, 14.19475998506890, 14.85044013693951, 14.99087358905120, 15.67076595921886

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