Properties

Label 2-4560-1.1-c1-0-53
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 + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s + 19-s + 25-s − 27-s + 6·29-s + 4·33-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s + 6·53-s − 4·55-s − 57-s + 12·59-s − 2·61-s − 2·65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 0.132·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.951709310969720145903036460751, −7.18588778480643576214599951747, −6.48546444924631199286740313682, −5.65472735320581306164695601367, −5.10907989527230163368059071437, −4.44653468539808003273181891597, −3.20436295081378722864928348772, −2.46088499191338758874187664358, −1.31864864232970058398235199804, 0, 1.31864864232970058398235199804, 2.46088499191338758874187664358, 3.20436295081378722864928348772, 4.44653468539808003273181891597, 5.10907989527230163368059071437, 5.65472735320581306164695601367, 6.48546444924631199286740313682, 7.18588778480643576214599951747, 7.951709310969720145903036460751

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