Properties

Label 2-585-1.1-c1-0-16
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  − 2·4-s + 5-s − 7-s − 3·11-s + 13-s + 4·16-s − 3·17-s − 4·19-s − 2·20-s − 9·23-s + 25-s + 2·28-s − 6·29-s + 2·31-s − 35-s − 37-s − 3·41-s + 2·43-s + 6·44-s − 6·47-s − 6·49-s − 2·52-s + 9·53-s − 3·55-s − 12·59-s + 5·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.277·13-s + 16-s − 0.727·17-s − 0.917·19-s − 0.447·20-s − 1.87·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s + 0.359·31-s − 0.169·35-s − 0.164·37-s − 0.468·41-s + 0.304·43-s + 0.904·44-s − 0.875·47-s − 6/7·49-s − 0.277·52-s + 1.23·53-s − 0.404·55-s − 1.56·59-s + 0.640·61-s − 64-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 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−10.14051697604131672798069750399, −9.458524366082460420320492630341, −8.532147401344436226602590366692, −7.83420991105047635513056919982, −6.46026292798114514724274972377, −5.63142717562375152012516639837, −4.60785132555940900382676705301, −3.60408487766035450933813281986, −2.10991322139674442815207973110, 0, 2.10991322139674442815207973110, 3.60408487766035450933813281986, 4.60785132555940900382676705301, 5.63142717562375152012516639837, 6.46026292798114514724274972377, 7.83420991105047635513056919982, 8.532147401344436226602590366692, 9.458524366082460420320492630341, 10.14051697604131672798069750399

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