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Properties

Label 2-139650-1.1-c1-0-12
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 − 11-s − 12-s − 4·13-s + 16-s + 2·17-s + 18-s − 19-s − 22-s − 2·23-s − 24-s − 4·26-s − 27-s + 5·29-s − 5·31-s + 32-s + 33-s + 2·34-s + 36-s + 6·37-s − 38-s + 4·39-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 − 0.301·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.229·19-s − 0.213·22-s − 0.417·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.928·29-s − 0.898·31-s + 0.176·32-s + 0.174·33-s + 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.162·38-s + 0.640·39-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

\(L(1)\) \(\approx\) \(1.483358889\)
\(L(\frac12)\) \(\approx\) \(1.483358889\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.27724280861945, −12.86701686169014, −12.50387565133114, −12.01225523234601, −11.63020521893778, −11.08819276384219, −10.65993157812278, −10.06021446973439, −9.710587822363597, −9.254243337841629, −8.297394637495901, −7.971491965230399, −7.490127390657308, −6.755676453687450, −6.556563001689977, −5.865999140405612, −5.300053316010899, −4.981147960501380, −4.396735486199498, −3.904417165662007, −3.072790740794308, −2.727693668910354, −1.894401355002903, −1.358935469474041, −0.3233841675508071, 0.3233841675508071, 1.358935469474041, 1.894401355002903, 2.727693668910354, 3.072790740794308, 3.904417165662007, 4.396735486199498, 4.981147960501380, 5.300053316010899, 5.865999140405612, 6.556563001689977, 6.755676453687450, 7.490127390657308, 7.971491965230399, 8.297394637495901, 9.254243337841629, 9.710587822363597, 10.06021446973439, 10.65993157812278, 11.08819276384219, 11.63020521893778, 12.01225523234601, 12.50387565133114, 12.86701686169014, 13.27724280861945

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