Properties

Label 2-4560-1.1-c1-0-50
Degree $2$
Conductor $4560$
Sign $-1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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 + 2·7-s + 9-s − 4·13-s + 15-s − 2·17-s + 19-s − 2·21-s + 6·23-s + 25-s − 27-s − 2·29-s − 2·35-s − 4·37-s + 4·39-s + 2·41-s − 2·43-s − 45-s + 6·47-s − 3·49-s + 2·51-s − 12·53-s − 57-s + 4·59-s + 6·61-s + 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.229·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.338·35-s − 0.657·37-s + 0.640·39-s + 0.312·41-s − 0.304·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.64·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s + 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.81069973683552537764219633572, −7.26638470980080213201722549918, −6.64071348241599421036541597605, −5.62078747703675619671923600515, −4.91258465380057120474005063394, −4.47416950603366237031202564520, −3.39120538263897498986822023931, −2.37176213738933915431121152151, −1.28629106315012200211845850675, 0, 1.28629106315012200211845850675, 2.37176213738933915431121152151, 3.39120538263897498986822023931, 4.47416950603366237031202564520, 4.91258465380057120474005063394, 5.62078747703675619671923600515, 6.64071348241599421036541597605, 7.26638470980080213201722549918, 7.81069973683552537764219633572

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