Properties

Label 2-23520-1.1-c1-0-42
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 − 2·11-s + 4·13-s − 15-s + 2·17-s − 6·19-s + 4·23-s + 25-s + 27-s − 10·29-s + 2·31-s − 2·33-s − 2·37-s + 4·39-s + 10·41-s − 4·43-s − 45-s − 8·47-s + 2·51-s + 4·53-s + 2·55-s − 6·57-s + 4·59-s + 2·61-s − 4·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s + 0.485·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.359·31-s − 0.348·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s + 0.549·53-s + 0.269·55-s − 0.794·57-s + 0.520·59-s + 0.256·61-s − 0.496·65-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−15.57845883822778, −15.16915688750162, −14.69010231833712, −14.25454592023709, −13.29241442185343, −13.15759611564274, −12.71018646407432, −11.89008728451098, −11.29817684241992, −10.76142639291699, −10.39453634595089, −9.527163715328356, −9.046489579170611, −8.422492545761078, −8.035769088409884, −7.402762141108314, −6.800511053030912, −6.085425634890318, −5.471052899590203, −4.691940179745022, −3.977661618427589, −3.508485046729758, −2.758619827976275, −1.982855661932923, −1.135011283035165, 0, 1.135011283035165, 1.982855661932923, 2.758619827976275, 3.508485046729758, 3.977661618427589, 4.691940179745022, 5.471052899590203, 6.085425634890318, 6.800511053030912, 7.402762141108314, 8.035769088409884, 8.422492545761078, 9.046489579170611, 9.527163715328356, 10.39453634595089, 10.76142639291699, 11.29817684241992, 11.89008728451098, 12.71018646407432, 13.15759611564274, 13.29241442185343, 14.25454592023709, 14.69010231833712, 15.16915688750162, 15.57845883822778

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