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Properties

Label 2-139650-1.1-c1-0-140
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 6·11-s − 12-s + 16-s − 3·17-s − 18-s − 19-s − 6·22-s + 6·23-s + 24-s − 27-s − 8·29-s − 8·31-s − 32-s − 6·33-s + 3·34-s + 36-s − 4·37-s + 38-s + 2·41-s + 4·43-s + 6·44-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 + 1.80·11-s − 0.288·12-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s − 1.27·22-s + 1.25·23-s + 0.204·24-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.176·32-s − 1.04·33-s + 0.514·34-s + 1/6·36-s − 0.657·37-s + 0.162·38-s + 0.312·41-s + 0.609·43-s + 0.904·44-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: \(1\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−13.54944831371072, −13.09267612625088, −12.53162492506304, −12.20255581548751, −11.48816225269773, −11.26712474109741, −10.85878440651691, −10.41460174923012, −9.508817332216796, −9.345075531585711, −8.993005925451144, −8.439983452719808, −7.716020907851765, −7.122333002922039, −6.863171147415916, −6.367715521604704, −5.796457212527811, −5.265506329934148, −4.613736852014453, −3.872378301342607, −3.643755328341793, −2.753076161852927, −1.887130757527518, −1.538824893246896, −0.7904655380402325, 0, 0.7904655380402325, 1.538824893246896, 1.887130757527518, 2.753076161852927, 3.643755328341793, 3.872378301342607, 4.613736852014453, 5.265506329934148, 5.796457212527811, 6.367715521604704, 6.863171147415916, 7.122333002922039, 7.716020907851765, 8.439983452719808, 8.993005925451144, 9.345075531585711, 9.508817332216796, 10.41460174923012, 10.85878440651691, 11.26712474109741, 11.48816225269773, 12.20255581548751, 12.53162492506304, 13.09267612625088, 13.54944831371072

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