Properties

Label 2-139650-1.1-c1-0-101
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 5·11-s − 12-s − 3·13-s + 16-s − 6·17-s − 18-s − 19-s − 5·22-s + 23-s + 24-s + 3·26-s − 27-s − 6·29-s − 7·31-s − 32-s − 5·33-s + 6·34-s + 36-s + 6·37-s + 38-s + 3·39-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 + 1.50·11-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s − 1.06·22-s + 0.208·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 1.11·29-s − 1.25·31-s − 0.176·32-s − 0.870·33-s + 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.162·38-s + 0.480·39-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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.63515220787859, −12.92638163389414, −12.72154924533250, −12.07584304235096, −11.52164942764271, −11.33008591028032, −10.87032680309218, −10.24325271845979, −9.762080115873641, −9.166048160215933, −9.041860653344660, −8.422203969878518, −7.705062040709167, −7.150234630998207, −6.859596406221613, −6.358910310426707, −5.832513215089538, −5.198935685300526, −4.619186656040628, −3.961803802766199, −3.621431961140474, −2.614531419833358, −2.038007137082537, −1.543234701971718, −0.6952204384174193, 0, 0.6952204384174193, 1.543234701971718, 2.038007137082537, 2.614531419833358, 3.621431961140474, 3.961803802766199, 4.619186656040628, 5.198935685300526, 5.832513215089538, 6.358910310426707, 6.859596406221613, 7.150234630998207, 7.705062040709167, 8.422203969878518, 9.041860653344660, 9.166048160215933, 9.762080115873641, 10.24325271845979, 10.87032680309218, 11.33008591028032, 11.52164942764271, 12.07584304235096, 12.72154924533250, 12.92638163389414, 13.63515220787859

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