Properties

Label 2-139650-1.1-c1-0-14
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 + 2·11-s − 12-s − 4·13-s + 16-s + 3·17-s + 18-s + 19-s + 2·22-s − 2·23-s − 24-s − 4·26-s − 27-s − 8·31-s + 32-s − 2·33-s + 3·34-s + 36-s − 12·37-s + 38-s + 4·39-s − 6·41-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.426·22-s − 0.417·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 1.43·31-s + 0.176·32-s − 0.348·33-s + 0.514·34-s + 1/6·36-s − 1.97·37-s + 0.162·38-s + 0.640·39-s − 0.937·41-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

\(L(1)\) \(\approx\) \(1.731223650\)
\(L(\frac12)\) \(\approx\) \(1.731223650\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−13.56720276739684, −12.75653606659574, −12.32534401082838, −12.06121717728394, −11.78871865229175, −11.03543026895235, −10.54090757817065, −10.29663467627538, −9.509388911941326, −9.237646123806090, −8.552430211257550, −7.784032797447319, −7.393780659678474, −6.995449308473260, −6.414171356378884, −5.848125884701188, −5.392086136748151, −4.925709696602838, −4.443964682193049, −3.696778992084918, −3.362051891766192, −2.616426152957312, −1.787726167787715, −1.451357488537521, −0.3451752658052160, 0.3451752658052160, 1.451357488537521, 1.787726167787715, 2.616426152957312, 3.362051891766192, 3.696778992084918, 4.443964682193049, 4.925709696602838, 5.392086136748151, 5.848125884701188, 6.414171356378884, 6.995449308473260, 7.393780659678474, 7.784032797447319, 8.552430211257550, 9.237646123806090, 9.509388911941326, 10.29663467627538, 10.54090757817065, 11.03543026895235, 11.78871865229175, 12.06121717728394, 12.32534401082838, 12.75653606659574, 13.56720276739684

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