Properties

Label 2-23520-1.1-c1-0-22
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

−15.80785108872954, −15.23107821662168, −14.74511518562114, −14.14365887269232, −13.42480702189859, −13.03020954044254, −12.32910875105604, −11.98545937825038, −11.32788752298935, −10.90137437322746, −10.35002341261174, −9.687478569339392, −9.179195216642414, −8.369394832582831, −7.964723574342801, −7.276352098794111, −6.570140726441420, −6.214719550866048, −5.471946640824109, −4.659628160490358, −4.231756426348396, −3.687230850750082, −2.407657085985740, −2.157134661316001, −0.8207562090201125, 0, 0.8207562090201125, 2.157134661316001, 2.407657085985740, 3.687230850750082, 4.231756426348396, 4.659628160490358, 5.471946640824109, 6.214719550866048, 6.570140726441420, 7.276352098794111, 7.964723574342801, 8.369394832582831, 9.179195216642414, 9.687478569339392, 10.35002341261174, 10.90137437322746, 11.32788752298935, 11.98545937825038, 12.32910875105604, 13.03020954044254, 13.42480702189859, 14.14365887269232, 14.74511518562114, 15.23107821662168, 15.80785108872954

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