Properties

Label 2-585-1.1-c1-0-8
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 − 3·7-s + 2·10-s + 5·11-s + 13-s + 6·14-s − 4·16-s − 5·17-s + 2·19-s − 2·20-s − 10·22-s + 23-s + 25-s − 2·26-s − 6·28-s − 10·29-s − 2·31-s + 8·32-s + 10·34-s + 3·35-s − 3·37-s − 4·38-s + 9·41-s − 4·43-s + 10·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 1.13·7-s + 0.632·10-s + 1.50·11-s + 0.277·13-s + 1.60·14-s − 16-s − 1.21·17-s + 0.458·19-s − 0.447·20-s − 2.13·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 1.13·28-s − 1.85·29-s − 0.359·31-s + 1.41·32-s + 1.71·34-s + 0.507·35-s − 0.493·37-s − 0.648·38-s + 1.40·41-s − 0.609·43-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−9.975565540863668498254961727135, −9.175134422877607002063538220098, −8.923978077954925138616609003816, −7.66337536315553181125482042712, −6.88984608666809355423599409217, −6.15614460284795576397102652158, −4.40005354130592369090367498749, −3.33489564696074388776632712954, −1.62636499494931476921849119993, 0, 1.62636499494931476921849119993, 3.33489564696074388776632712954, 4.40005354130592369090367498749, 6.15614460284795576397102652158, 6.88984608666809355423599409217, 7.66337536315553181125482042712, 8.923978077954925138616609003816, 9.175134422877607002063538220098, 9.975565540863668498254961727135

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