Properties

Label 2-23520-1.1-c1-0-36
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 + 4·23-s + 25-s − 27-s − 2·29-s − 8·31-s + 6·37-s − 2·39-s + 6·41-s − 12·43-s − 45-s − 12·47-s − 6·51-s − 10·53-s + 8·59-s + 10·61-s − 2·65-s + 12·67-s − 4·69-s − 8·71-s − 10·73-s − 75-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 + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s − 0.149·45-s − 1.75·47-s − 0.840·51-s − 1.37·53-s + 1.04·59-s + 1.28·61-s − 0.248·65-s + 1.46·67-s − 0.481·69-s − 0.949·71-s − 1.17·73-s − 0.115·75-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 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.97786003328105, −15.02587210560577, −14.72477618637465, −14.30223192546002, −13.36715145936599, −12.87773533726518, −12.63079700957966, −11.70765988050576, −11.43128164607868, −10.98356588496325, −10.23943839172553, −9.747559326825511, −9.178361889426183, −8.372228374367733, −7.923219088837230, −7.280895926248607, −6.718678277382517, −6.048723603666450, −5.375397792742175, −4.978535635228911, −4.059764597701272, −3.511250323463933, −2.873864354844476, −1.695672871307937, −1.043234091926824, 0, 1.043234091926824, 1.695672871307937, 2.873864354844476, 3.511250323463933, 4.059764597701272, 4.978535635228911, 5.375397792742175, 6.048723603666450, 6.718678277382517, 7.280895926248607, 7.923219088837230, 8.372228374367733, 9.178361889426183, 9.747559326825511, 10.23943839172553, 10.98356588496325, 11.43128164607868, 11.70765988050576, 12.63079700957966, 12.87773533726518, 13.36715145936599, 14.30223192546002, 14.72477618637465, 15.02587210560577, 15.97786003328105

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