Properties

Label 2-164730-1.1-c1-0-9
Degree $2$
Conductor $164730$
Sign $1$
Analytic cond. $1315.37$
Root an. cond. $36.2681$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s + 15-s + 16-s + 18-s − 19-s + 20-s − 4·22-s − 4·23-s + 24-s + 25-s + 2·26-s + 27-s − 6·29-s + 30-s − 4·31-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164730\)    =    \(2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1315.37\)
Root analytic conductor: \(36.2681\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{164730} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 164730,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.711610455\)
\(L(\frac12)\) \(\approx\) \(3.711610455\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.36981442054086, −12.87210679665281, −12.60204435596100, −11.84888106806058, −11.35138217928223, −10.95453097759279, −10.35357702667586, −9.980358145006087, −9.521949779714386, −8.913148388727350, −8.197544489616301, −8.084880994325083, −7.403043061984757, −6.908335012227295, −6.270027656974338, −5.869468687174325, −5.301082995207037, −4.810546597390753, −4.295735912646020, −3.458129571749598, −3.296752884030072, −2.558957268326425, −1.821320554613053, −1.692391030049330, −0.4291032283857462, 0.4291032283857462, 1.692391030049330, 1.821320554613053, 2.558957268326425, 3.296752884030072, 3.458129571749598, 4.295735912646020, 4.810546597390753, 5.301082995207037, 5.869468687174325, 6.270027656974338, 6.908335012227295, 7.403043061984757, 8.084880994325083, 8.197544489616301, 8.913148388727350, 9.521949779714386, 9.980358145006087, 10.35357702667586, 10.95453097759279, 11.35138217928223, 11.84888106806058, 12.60204435596100, 12.87210679665281, 13.36981442054086

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