Properties

Label 2-139650-1.1-c1-0-115
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 + 3·11-s − 12-s − 2·13-s + 16-s − 6·17-s − 18-s − 19-s − 3·22-s + 3·23-s + 24-s + 2·26-s − 27-s + 3·29-s + 7·31-s − 32-s − 3·33-s + 6·34-s + 36-s + 2·37-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 + 0.904·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.229·19-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.557·29-s + 1.25·31-s − 0.176·32-s − 0.522·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−13.53171239093580, −13.12766027207130, −12.56772080651601, −12.13862915049573, −11.53756567308638, −11.27093832498869, −10.90042740052875, −10.15389278704919, −9.843412284798172, −9.347440210842963, −8.842900127609664, −8.311352836853706, −7.924798503831522, −7.041236423973792, −6.845577559559684, −6.338536158101243, −5.953191053987818, −5.060957059031496, −4.582453038371817, −4.259380460858263, −3.281654737831406, −2.820606429800627, −2.000866883585809, −1.500545648185706, −0.7232805192097839, 0, 0.7232805192097839, 1.500545648185706, 2.000866883585809, 2.820606429800627, 3.281654737831406, 4.259380460858263, 4.582453038371817, 5.060957059031496, 5.953191053987818, 6.338536158101243, 6.845577559559684, 7.041236423973792, 7.924798503831522, 8.311352836853706, 8.842900127609664, 9.347440210842963, 9.843412284798172, 10.15389278704919, 10.90042740052875, 11.27093832498869, 11.53756567308638, 12.13862915049573, 12.56772080651601, 13.12766027207130, 13.53171239093580

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