Properties

Label 2-23520-1.1-c1-0-20
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 − 6·11-s − 4·13-s + 15-s − 6·17-s + 6·19-s + 4·23-s + 25-s − 27-s − 2·29-s − 2·31-s + 6·33-s + 6·37-s + 4·39-s − 6·41-s + 12·43-s − 45-s + 6·51-s − 4·53-s + 6·55-s − 6·57-s − 4·59-s + 10·61-s + 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 1.04·33-s + 0.986·37-s + 0.640·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s + 0.840·51-s − 0.549·53-s + 0.809·55-s − 0.794·57-s − 0.520·59-s + 1.28·61-s + 0.496·65-s + 1.46·67-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: Trivial
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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.61396556713147, −15.51142156419772, −14.68709798609492, −14.15850839704161, −13.34062188168895, −12.91599213518486, −12.66662767330566, −11.74633135376231, −11.42426141566537, −10.83681165671731, −10.37232193475766, −9.701762447351764, −9.208875251106237, −8.416988881376152, −7.771164396081124, −7.241906381212852, −6.969088108882599, −5.879854949813932, −5.438915485328004, −4.756992065941002, −4.448129274732225, −3.331724936782141, −2.688394519765844, −2.061494212271190, −0.7738017414448675, 0, 0.7738017414448675, 2.061494212271190, 2.688394519765844, 3.331724936782141, 4.448129274732225, 4.756992065941002, 5.438915485328004, 5.879854949813932, 6.969088108882599, 7.241906381212852, 7.771164396081124, 8.416988881376152, 9.208875251106237, 9.701762447351764, 10.37232193475766, 10.83681165671731, 11.42426141566537, 11.74633135376231, 12.66662767330566, 12.91599213518486, 13.34062188168895, 14.15850839704161, 14.68709798609492, 15.51142156419772, 15.61396556713147

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