Properties

Label 2-73920-1.1-c1-0-110
Degree $2$
Conductor $73920$
Sign $-1$
Analytic cond. $590.254$
Root an. cond. $24.2951$
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 + 7-s + 9-s + 11-s + 2·13-s + 15-s + 2·17-s − 4·19-s − 21-s + 25-s − 27-s − 6·29-s − 33-s − 35-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s − 2·51-s + 2·53-s − 55-s + 4·57-s − 4·59-s − 6·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.169·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 0.274·53-s − 0.134·55-s + 0.529·57-s − 0.520·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73920\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(590.254\)
Root analytic conductor: \(24.2951\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{73920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 73920,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.43233356705557, −13.64025255832770, −13.47504362550976, −12.64265308388663, −12.31626805036720, −11.84523750182849, −11.31097796546009, −10.85239865892560, −10.48366569470453, −9.910984175077186, −9.120406417363170, −8.833102698697115, −8.162497310719921, −7.629661963047545, −7.180303937150532, −6.480979062521459, −6.069006179851316, −5.458560687060643, −4.847241707860054, −4.352531696480751, −3.641468761402151, −3.295234734083726, −2.179986856431574, −1.656366669402769, −0.8438458443004797, 0, 0.8438458443004797, 1.656366669402769, 2.179986856431574, 3.295234734083726, 3.641468761402151, 4.352531696480751, 4.847241707860054, 5.458560687060643, 6.069006179851316, 6.480979062521459, 7.180303937150532, 7.629661963047545, 8.162497310719921, 8.833102698697115, 9.120406417363170, 9.910984175077186, 10.48366569470453, 10.85239865892560, 11.31097796546009, 11.84523750182849, 12.31626805036720, 12.64265308388663, 13.47504362550976, 13.64025255832770, 14.43233356705557

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