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Properties

Label 2-139650-1.1-c1-0-148
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 + 2·13-s + 16-s + 3·17-s + 18-s − 19-s + 4·22-s + 2·23-s + 24-s + 2·26-s + 27-s + 2·29-s − 8·31-s + 32-s + 4·33-s + 3·34-s + 36-s + 10·37-s − 38-s + 2·39-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 + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s + 0.417·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 0.514·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s + 0.320·39-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

\(L(1)\) \(\approx\) \(8.428316541\)
\(L(\frac12)\) \(\approx\) \(8.428316541\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−13.32979870961291, −12.96289482788510, −12.63201757256406, −12.06562907215758, −11.49343168946936, −11.12638850935979, −10.69531874738062, −9.982042287417313, −9.430307047684593, −9.162815748812377, −8.563688440198624, −7.893003905160505, −7.600329846510486, −6.924445957146049, −6.421643851294684, −6.027266837905050, −5.389710098232622, −4.832485033445955, −4.038343030889935, −3.866824602013003, −3.320636678621872, −2.549521342590278, −2.107095132002220, −1.213998121808412, −0.8278471509590393, 0.8278471509590393, 1.213998121808412, 2.107095132002220, 2.549521342590278, 3.320636678621872, 3.866824602013003, 4.038343030889935, 4.832485033445955, 5.389710098232622, 6.027266837905050, 6.421643851294684, 6.924445957146049, 7.600329846510486, 7.893003905160505, 8.563688440198624, 9.162815748812377, 9.430307047684593, 9.982042287417313, 10.69531874738062, 11.12638850935979, 11.49343168946936, 12.06562907215758, 12.63201757256406, 12.96289482788510, 13.32979870961291

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