Properties

Label 2-23520-1.1-c1-0-28
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 − 8·23-s + 25-s − 27-s + 10·29-s − 4·31-s − 4·33-s − 6·37-s + 6·39-s − 6·41-s + 4·43-s − 45-s + 12·47-s − 6·51-s + 6·53-s − 4·55-s + 4·57-s − 4·59-s + 2·61-s + 6·65-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 − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s − 0.696·33-s − 0.986·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.744·65-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 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.73170743745369, −15.19610878519448, −14.57375809675465, −14.16264752782792, −13.80703759839736, −12.63689934780069, −12.31752461942886, −11.99810710131623, −11.66430183387009, −10.69048912718428, −10.22884348251405, −9.855849357045859, −9.137507661311943, −8.452764603678448, −7.868299950854181, −7.202825110918281, −6.787673696750643, −6.058420595587918, −5.474336546802319, −4.769710019762561, −4.143887171826016, −3.640516453964625, −2.663462171942924, −1.876608254086725, −0.9388095236755851, 0, 0.9388095236755851, 1.876608254086725, 2.663462171942924, 3.640516453964625, 4.143887171826016, 4.769710019762561, 5.474336546802319, 6.058420595587918, 6.787673696750643, 7.202825110918281, 7.868299950854181, 8.452764603678448, 9.137507661311943, 9.855849357045859, 10.22884348251405, 10.69048912718428, 11.66430183387009, 11.99810710131623, 12.31752461942886, 12.63689934780069, 13.80703759839736, 14.16264752782792, 14.57375809675465, 15.19610878519448, 15.73170743745369

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