Properties

Label 2-139650-1.1-c1-0-106
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 + 4·11-s + 12-s − 4·13-s + 16-s + 6·17-s + 18-s + 19-s + 4·22-s − 8·23-s + 24-s − 4·26-s + 27-s + 2·29-s + 8·31-s + 32-s + 4·33-s + 6·34-s + 36-s + 8·37-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 + 1.20·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-s + 1.31·37-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: $\chi_{139650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.133426044\)
\(L(\frac12)\) \(\approx\) \(7.133426044\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.64154319552628, −12.81974302168718, −12.43789110967003, −12.03841678371237, −11.80019439224091, −11.12087560023609, −10.49113767106654, −9.933480394573293, −9.538986243344752, −9.312350089020044, −8.362578058719857, −7.866690586371949, −7.689980437564377, −6.953192273880760, −6.463784635671508, −5.872776140020839, −5.539086483609718, −4.646897912022761, −4.231264243917782, −3.923182640692611, −3.046298973585914, −2.735681009393215, −2.057797930042949, −1.310606705208985, −0.7095740187318204, 0.7095740187318204, 1.310606705208985, 2.057797930042949, 2.735681009393215, 3.046298973585914, 3.923182640692611, 4.231264243917782, 4.646897912022761, 5.539086483609718, 5.872776140020839, 6.463784635671508, 6.953192273880760, 7.689980437564377, 7.866690586371949, 8.362578058719857, 9.312350089020044, 9.538986243344752, 9.933480394573293, 10.49113767106654, 11.12087560023609, 11.80019439224091, 12.03841678371237, 12.43789110967003, 12.81974302168718, 13.64154319552628

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