Properties

Label 2-139650-1.1-c1-0-105
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 + 6·11-s − 12-s + 16-s + 7·17-s + 18-s − 19-s + 6·22-s + 6·23-s − 24-s − 27-s − 4·29-s + 8·31-s + 32-s − 6·33-s + 7·34-s + 36-s + 8·37-s − 38-s − 10·41-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 + 1.69·17-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 − 0.742·29-s + 1.43·31-s + 0.176·32-s − 1.04·33-s + 1.20·34-s + 1/6·36-s + 1.31·37-s − 0.162·38-s − 1.56·41-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: \(0\)
Selberg data: \((2,\ 139650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.379128596\)
\(L(\frac12)\) \(\approx\) \(5.379128596\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.24201079889612, −13.10266887949457, −12.24570042489373, −11.98221331448762, −11.63162029526996, −11.32246246875700, −10.52324365593308, −10.10191958134812, −9.749854911711931, −8.936444016830404, −8.732397098095685, −7.833535395008192, −7.396090499326210, −6.876442110483498, −6.368215795410473, −5.977068611880498, −5.467611854647063, −4.795845372175042, −4.430878430899003, −3.732703840541345, −3.333508130709373, −2.724706422864139, −1.742391026177852, −1.239702592185978, −0.7079398798325962, 0.7079398798325962, 1.239702592185978, 1.742391026177852, 2.724706422864139, 3.333508130709373, 3.732703840541345, 4.430878430899003, 4.795845372175042, 5.467611854647063, 5.977068611880498, 6.368215795410473, 6.876442110483498, 7.396090499326210, 7.833535395008192, 8.732397098095685, 8.936444016830404, 9.749854911711931, 10.10191958134812, 10.52324365593308, 11.32246246875700, 11.63162029526996, 11.98221331448762, 12.24570042489373, 13.10266887949457, 13.24201079889612

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