Properties

Label 2-189618-1.1-c1-0-9
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 − 2·5-s − 6-s − 4·7-s − 8-s + 9-s + 2·10-s + 11-s + 12-s + 4·14-s − 2·15-s + 16-s + 17-s − 18-s + 4·19-s − 2·20-s − 4·21-s − 22-s − 4·23-s − 24-s − 25-s + 27-s − 4·28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.872·21-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.365·30-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

\(L(1)\) \(\approx\) \(1.230883582\)
\(L(\frac12)\) \(\approx\) \(1.230883582\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−12.97033230591003, −12.68775470429601, −12.03597976264390, −11.70620421625248, −11.40826248280426, −10.43139402176500, −10.26138456360258, −9.731636959628336, −9.301594767724680, −8.955600048264292, −8.317770146234721, −7.789701187855743, −7.551711988092390, −6.969600994624235, −6.419658694517495, −6.038019871734625, −5.439598901958467, −4.472013037632503, −4.096478659445665, −3.458579801429504, −3.069931858231040, −2.609027211528101, −1.811424671628622, −0.9659723863428163, −0.4088914034603153, 0.4088914034603153, 0.9659723863428163, 1.811424671628622, 2.609027211528101, 3.069931858231040, 3.458579801429504, 4.096478659445665, 4.472013037632503, 5.439598901958467, 6.038019871734625, 6.419658694517495, 6.969600994624235, 7.551711988092390, 7.789701187855743, 8.317770146234721, 8.955600048264292, 9.301594767724680, 9.731636959628336, 10.26138456360258, 10.43139402176500, 11.40826248280426, 11.70620421625248, 12.03597976264390, 12.68775470429601, 12.97033230591003

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