Properties

Label 2-4560-1.1-c1-0-70
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 − 2·13-s + 15-s − 6·17-s − 19-s − 4·23-s + 25-s + 27-s − 6·29-s − 2·37-s − 2·39-s − 2·41-s + 4·43-s + 45-s + 4·47-s − 7·49-s − 6·51-s − 6·53-s − 57-s − 12·59-s − 2·61-s − 2·65-s − 4·67-s − 4·69-s − 6·73-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 − 0.229·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s − 49-s − 0.840·51-s − 0.824·53-s − 0.132·57-s − 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.481·69-s − 0.702·73-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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−7.940697392568063894516646881588, −7.33130169417677965919759318765, −6.51684547020630583970980341182, −5.87610561072860476159412569629, −4.85605278362403563566646396128, −4.23903621700509192415534598739, −3.27009919545542356583152960436, −2.32788430305649778783254846326, −1.69602716497593410840434651624, 0, 1.69602716497593410840434651624, 2.32788430305649778783254846326, 3.27009919545542356583152960436, 4.23903621700509192415534598739, 4.85605278362403563566646396128, 5.87610561072860476159412569629, 6.51684547020630583970980341182, 7.33130169417677965919759318765, 7.940697392568063894516646881588

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