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Properties

Label 2-139650-1.1-c1-0-131
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 + 3·11-s + 12-s + 5·13-s + 16-s + 18-s − 19-s + 3·22-s + 9·23-s + 24-s + 5·26-s + 27-s − 5·31-s + 32-s + 3·33-s + 36-s − 8·37-s − 38-s + 5·39-s + 6·41-s − 5·43-s + 3·44-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 + 0.904·11-s + 0.288·12-s + 1.38·13-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.639·22-s + 1.87·23-s + 0.204·24-s + 0.980·26-s + 0.192·27-s − 0.898·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.31·37-s − 0.162·38-s + 0.800·39-s + 0.937·41-s − 0.762·43-s + 0.452·44-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

\(L(1)\) \(\approx\) \(8.077550629\)
\(L(\frac12)\) \(\approx\) \(8.077550629\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.34048322022777, −13.05904065859608, −12.56203539201389, −12.15289697599911, −11.40100461750064, −11.01975540476104, −10.82680488585410, −10.04748976204264, −9.436982961637241, −8.989271347819682, −8.595890239630735, −8.136761259753002, −7.346554684782366, −6.939794591513162, −6.561904033887497, −5.940880317026895, −5.377150641571421, −4.854605715235395, −4.173893806538848, −3.567113928048779, −3.478818096915667, −2.629498640002073, −1.980538191013567, −1.318058200569284, −0.7708325917272709, 0.7708325917272709, 1.318058200569284, 1.980538191013567, 2.629498640002073, 3.478818096915667, 3.567113928048779, 4.173893806538848, 4.854605715235395, 5.377150641571421, 5.940880317026895, 6.561904033887497, 6.939794591513162, 7.346554684782366, 8.136761259753002, 8.595890239630735, 8.989271347819682, 9.436982961637241, 10.04748976204264, 10.82680488585410, 11.01975540476104, 11.40100461750064, 12.15289697599911, 12.56203539201389, 13.05904065859608, 13.34048322022777

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