Properties

Label 2-585-1.1-c1-0-17
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-s − 4-s − 5-s − 3·8-s − 10-s − 4·11-s + 13-s − 16-s − 2·17-s − 4·19-s + 20-s − 4·22-s − 8·23-s + 25-s + 26-s + 2·29-s − 8·31-s + 5·32-s − 2·34-s + 6·37-s − 4·38-s + 3·40-s + 6·41-s − 4·43-s + 4·44-s − 8·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.986·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + 1.16·47-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 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−10.36766611156745383388084599979, −9.359576043548733628235969266343, −8.404296066573383401625737694753, −7.73233535694260719129401432761, −6.37803160504271454727036453417, −5.52200307165618101356281230656, −4.51284738641056261325871513649, −3.73104630189750446831233920392, −2.43381471255309129843962850009, 0, 2.43381471255309129843962850009, 3.73104630189750446831233920392, 4.51284738641056261325871513649, 5.52200307165618101356281230656, 6.37803160504271454727036453417, 7.73233535694260719129401432761, 8.404296066573383401625737694753, 9.359576043548733628235969266343, 10.36766611156745383388084599979

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