Properties

Label 2-53130-1.1-c1-0-43
Degree $2$
Conductor $53130$
Sign $-1$
Analytic cond. $424.245$
Root an. cond. $20.5972$
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 + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 11-s + 12-s + 6·13-s + 14-s − 15-s + 16-s − 2·17-s − 18-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s − 6·26-s + 27-s − 28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53130\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 23\)
Sign: $-1$
Analytic conductor: \(424.245\)
Root analytic conductor: \(20.5972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 53130,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
23 \( 1 - T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.85851975088344, −14.08686717643673, −13.79595288079330, −13.08917523588275, −12.74031731847148, −12.20687588694498, −11.34277143603575, −11.12933332375421, −10.58550629492981, −10.04914606311492, −9.397360381406609, −8.836065218177585, −8.597911648063146, −8.033794151718464, −7.428050864055061, −6.850851082501031, −6.472842133569250, −5.650674125247170, −5.149386413166689, −4.154688405033492, −3.560059569386858, −3.314319404566226, −2.315431574744857, −1.746648186177031, −0.9218414373953447, 0, 0.9218414373953447, 1.746648186177031, 2.315431574744857, 3.314319404566226, 3.560059569386858, 4.154688405033492, 5.149386413166689, 5.650674125247170, 6.472842133569250, 6.850851082501031, 7.428050864055061, 8.033794151718464, 8.597911648063146, 8.836065218177585, 9.397360381406609, 10.04914606311492, 10.58550629492981, 11.12933332375421, 11.34277143603575, 12.20687588694498, 12.74031731847148, 13.08917523588275, 13.79595288079330, 14.08686717643673, 14.85851975088344

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