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Properties

Label 2-139650-1.1-c1-0-107
Degree $2$
Conductor $139650$
Sign $1$
Analytic cond. $1115.11$
Root an. cond. $33.3932$
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 + 6-s + 8-s + 9-s + 4·11-s + 12-s − 2·13-s + 16-s − 2·17-s + 18-s + 19-s + 4·22-s + 6·23-s + 24-s − 2·26-s + 27-s + 6·29-s + 6·31-s + 32-s + 4·33-s − 2·34-s + 36-s − 10·37-s + 38-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.852·22-s + 1.25·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s − 1.64·37-s + 0.162·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(139650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(1115.11\)
Root analytic conductor: \(33.3932\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.158850416\)
\(L(\frac12)\) \(\approx\) \(7.158850416\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
19 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.61881715699055, −12.94194691515416, −12.47564215834478, −12.03615386102725, −11.68636478391206, −11.09562680513882, −10.56255485485125, −10.02282931700028, −9.597670320412152, −8.894751083473589, −8.661857945655760, −8.102189035429176, −7.228886357682242, −7.054966690387704, −6.570767821757736, −5.994612551267197, −5.297858362110101, −4.729710433739060, −4.382314060627000, −3.689393290523177, −3.234981034519311, −2.622797909644917, −2.083552146320387, −1.307682084527631, −0.7112833029643810, 0.7112833029643810, 1.307682084527631, 2.083552146320387, 2.622797909644917, 3.234981034519311, 3.689393290523177, 4.382314060627000, 4.729710433739060, 5.297858362110101, 5.994612551267197, 6.570767821757736, 7.054966690387704, 7.228886357682242, 8.102189035429176, 8.661857945655760, 8.894751083473589, 9.597670320412152, 10.02282931700028, 10.56255485485125, 11.09562680513882, 11.68636478391206, 12.03615386102725, 12.47564215834478, 12.94194691515416, 13.61881715699055

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