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Properties

Label 2-53550-1.1-c1-0-115
Degree $2$
Conductor $53550$
Sign $-1$
Analytic cond. $427.598$
Root an. cond. $20.6784$
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 + 7-s + 8-s + 6·13-s + 14-s + 16-s + 17-s − 8·23-s + 6·26-s + 28-s + 6·29-s − 8·31-s + 32-s + 34-s − 10·37-s + 6·41-s − 12·43-s − 8·46-s + 49-s + 6·52-s − 10·53-s + 56-s + 6·58-s + 8·59-s + 6·61-s − 8·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.66·23-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s − 1.64·37-s + 0.937·41-s − 1.82·43-s − 1.17·46-s + 1/7·49-s + 0.832·52-s − 1.37·53-s + 0.133·56-s + 0.787·58-s + 1.04·59-s + 0.768·61-s − 1.01·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(427.598\)
Root analytic conductor: \(20.6784\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53550,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
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

−14.62151085906488, −13.98931201669038, −13.86800910551081, −13.19827761314271, −12.75116855774486, −12.09158857249845, −11.74728980143576, −11.18417590812577, −10.65079714766522, −10.26525141152989, −9.605507370300903, −8.813948350131033, −8.415434173502522, −7.914243416625773, −7.297946909473797, −6.513526353483933, −6.246452490122872, −5.543534436783770, −5.095578488172456, −4.358213402309260, −3.640406472230084, −3.501356498529369, −2.482924123736743, −1.730347004354000, −1.248368364150141, 0, 1.248368364150141, 1.730347004354000, 2.482924123736743, 3.501356498529369, 3.640406472230084, 4.358213402309260, 5.095578488172456, 5.543534436783770, 6.246452490122872, 6.513526353483933, 7.297946909473797, 7.914243416625773, 8.415434173502522, 8.813948350131033, 9.605507370300903, 10.26525141152989, 10.65079714766522, 11.18417590812577, 11.74728980143576, 12.09158857249845, 12.75116855774486, 13.19827761314271, 13.86800910551081, 13.98931201669038, 14.62151085906488

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