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Properties

Label 2-139650-1.1-c1-0-199
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 + 3·11-s + 12-s − 4·13-s + 16-s − 4·17-s + 18-s + 19-s + 3·22-s + 4·23-s + 24-s − 4·26-s + 27-s − 7·29-s − 7·31-s + 32-s + 3·33-s − 4·34-s + 36-s + 4·37-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 + 0.904·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s + 0.639·22-s + 0.834·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 1.29·29-s − 1.25·31-s + 0.176·32-s + 0.522·33-s − 0.685·34-s + 1/6·36-s + 0.657·37-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.75652628348889, −13.15408505571244, −12.73692771251374, −12.36558256872161, −11.83757584342327, −11.27131984883186, −10.88917891141336, −10.42112979355245, −9.574376630678587, −9.244662540617642, −9.067465089774543, −8.194984796221877, −7.653253294811203, −7.238855752865790, −6.773613881486696, −6.287159941826725, −5.620997892341959, −4.969284406977622, −4.663900458552448, −3.929207925204554, −3.533011450898033, −2.959269642614061, −2.149663823753367, −1.923682991678501, −1.010136399915791, 0, 1.010136399915791, 1.923682991678501, 2.149663823753367, 2.959269642614061, 3.533011450898033, 3.929207925204554, 4.663900458552448, 4.969284406977622, 5.620997892341959, 6.287159941826725, 6.773613881486696, 7.238855752865790, 7.653253294811203, 8.194984796221877, 9.067465089774543, 9.244662540617642, 9.574376630678587, 10.42112979355245, 10.88917891141336, 11.27131984883186, 11.83757584342327, 12.36558256872161, 12.73692771251374, 13.15408505571244, 13.75652628348889

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