Properties

Label 2-585-1.1-c1-0-18
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 − 4·7-s − 3·8-s + 10-s − 2·11-s − 13-s − 4·14-s − 16-s − 2·17-s − 6·19-s − 20-s − 2·22-s + 6·23-s + 25-s − 26-s + 4·28-s − 2·29-s − 10·31-s + 5·32-s − 2·34-s − 4·35-s − 2·37-s − 6·38-s − 3·40-s + 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s − 1.06·8-s + 0.316·10-s − 0.603·11-s − 0.277·13-s − 1.06·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 0.371·29-s − 1.79·31-s + 0.883·32-s − 0.342·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s − 0.474·40-s + 0.937·41-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−10.24076128195437772789084342069, −9.238371497422229278947323218386, −8.903866275348653436076253219750, −7.36945570129601409328488904448, −6.35831289645710659676663930685, −5.66957165960173077695586779200, −4.59731852050444182415870036677, −3.51994456797006266949922249859, −2.53409816778618263405222233425, 0, 2.53409816778618263405222233425, 3.51994456797006266949922249859, 4.59731852050444182415870036677, 5.66957165960173077695586779200, 6.35831289645710659676663930685, 7.36945570129601409328488904448, 8.903866275348653436076253219750, 9.238371497422229278947323218386, 10.24076128195437772789084342069

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