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Properties

Label 2-182070-1.1-c1-0-71
Degree $2$
Conductor $182070$
Sign $-1$
Analytic cond. $1453.83$
Root an. cond. $38.1292$
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 + 7-s − 8-s + 10-s + 11-s + 13-s − 14-s + 16-s − 20-s − 22-s + 4·23-s + 25-s − 26-s + 28-s + 10·29-s − 3·31-s − 32-s − 35-s − 2·37-s + 40-s − 5·41-s − 12·43-s + 44-s − 4·46-s + 3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.85·29-s − 0.538·31-s − 0.176·32-s − 0.169·35-s − 0.328·37-s + 0.158·40-s − 0.780·41-s − 1.82·43-s + 0.150·44-s − 0.589·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1453.83\)
Root analytic conductor: \(38.1292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 182070,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
17 \( 1 \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.23837559022122, −13.00963821473362, −12.19638972551178, −11.79618451924943, −11.66323304170481, −10.89706525616614, −10.57820060342167, −10.13448461716032, −9.579954368182540, −8.985284241177146, −8.573798111475024, −8.261991506751603, −7.721353483525156, −7.104966730916261, −6.736791495362506, −6.305160653107090, −5.582973604773890, −4.929527642734258, −4.651530331786610, −3.760027732074895, −3.375042926698130, −2.735005388434767, −2.053768461008851, −1.358778757126338, −0.8535642131508429, 0, 0.8535642131508429, 1.358778757126338, 2.053768461008851, 2.735005388434767, 3.375042926698130, 3.760027732074895, 4.651530331786610, 4.929527642734258, 5.582973604773890, 6.305160653107090, 6.736791495362506, 7.104966730916261, 7.721353483525156, 8.261991506751603, 8.573798111475024, 8.985284241177146, 9.579954368182540, 10.13448461716032, 10.57820060342167, 10.89706525616614, 11.66323304170481, 11.79618451924943, 12.19638972551178, 13.00963821473362, 13.23837559022122

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