Properties

Label 2-585-1.1-c1-0-11
Degree $2$
Conductor $585$
Sign $-1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s − 5·11-s − 13-s + 2·14-s − 4·16-s + 7·17-s − 6·19-s + 2·20-s + 10·22-s − 3·23-s + 25-s + 2·26-s − 2·28-s − 2·29-s + 2·31-s + 8·32-s − 14·34-s − 35-s + 7·37-s + 12·38-s − 9·41-s − 8·43-s − 10·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s − 1.50·11-s − 0.277·13-s + 0.534·14-s − 16-s + 1.69·17-s − 1.37·19-s + 0.447·20-s + 2.13·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.371·29-s + 0.359·31-s + 1.41·32-s − 2.40·34-s − 0.169·35-s + 1.15·37-s + 1.94·38-s − 1.40·41-s − 1.21·43-s − 1.50·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−10.03210824623968341800708629459, −9.664619125780354113128944873936, −8.333107995317450905784282789603, −7.997850874518894594264893149223, −6.93279974611194334514178729187, −5.86662536743360060967332495720, −4.75380031772719504345197730313, −3.02709857296590648239002053477, −1.78483850539275904424541360817, 0, 1.78483850539275904424541360817, 3.02709857296590648239002053477, 4.75380031772719504345197730313, 5.86662536743360060967332495720, 6.93279974611194334514178729187, 7.997850874518894594264893149223, 8.333107995317450905784282789603, 9.664619125780354113128944873936, 10.03210824623968341800708629459

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