Properties

Label 2-189618-1.1-c1-0-16
Degree $2$
Conductor $189618$
Sign $1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s + 4·14-s + 16-s + 17-s + 18-s + 2·19-s − 4·21-s + 22-s − 2·23-s − 24-s − 5·25-s − 27-s + 4·28-s + 6·29-s + 8·31-s + 32-s − 33-s + 34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.213·22-s − 0.417·23-s − 0.204·24-s − 25-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.808460084\)
\(L(\frac12)\) \(\approx\) \(4.808460084\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−13.06002805181947, −12.58854111951012, −11.98034422704853, −11.75044124337114, −11.47447775071359, −10.87212922063283, −10.44627347804306, −9.943642310607082, −9.462537617028144, −8.690140742309173, −8.176027270033989, −7.817517949030372, −7.381547294455013, −6.663598934033301, −6.229924168864202, −5.792322921402719, −5.075068513181853, −4.872694746959943, −4.323100121838043, −3.857652488547443, −3.085758977305503, −2.504690002864019, −1.660753643552355, −1.418472307575732, −0.5756125945085334, 0.5756125945085334, 1.418472307575732, 1.660753643552355, 2.504690002864019, 3.085758977305503, 3.857652488547443, 4.323100121838043, 4.872694746959943, 5.075068513181853, 5.792322921402719, 6.229924168864202, 6.663598934033301, 7.381547294455013, 7.817517949030372, 8.176027270033989, 8.690140742309173, 9.462537617028144, 9.943642310607082, 10.44627347804306, 10.87212922063283, 11.47447775071359, 11.75044124337114, 11.98034422704853, 12.58854111951012, 13.06002805181947

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