Properties

Label 2-189618-1.1-c1-0-45
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 $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 14-s − 15-s + 16-s − 17-s + 18-s − 8·19-s − 20-s − 21-s + 22-s − 7·23-s + 24-s − 4·25-s + 27-s − 28-s − 7·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s − 1.29·29-s − 0.182·30-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: \(2\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

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 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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

−13.50210583615799, −13.14144813682819, −12.65714994642131, −12.36110770336784, −11.80877103928289, −11.17808082712638, −10.98008554821149, −10.27552947406773, −9.806541640416631, −9.386071639393786, −8.744844100777751, −8.187631147114751, −7.992964230932642, −7.242334795763366, −6.811285903378652, −6.314969462438851, −5.890014427605881, −5.204283754614694, −4.613690007037370, −4.060405206097186, −3.590208353429960, −3.412893837719415, −2.427192676493090, −1.877227887313606, −1.632237291918871, 0, 0, 1.632237291918871, 1.877227887313606, 2.427192676493090, 3.412893837719415, 3.590208353429960, 4.060405206097186, 4.613690007037370, 5.204283754614694, 5.890014427605881, 6.314969462438851, 6.811285903378652, 7.242334795763366, 7.992964230932642, 8.187631147114751, 8.744844100777751, 9.386071639393786, 9.806541640416631, 10.27552947406773, 10.98008554821149, 11.17808082712638, 11.80877103928289, 12.36110770336784, 12.65714994642131, 13.14144813682819, 13.50210583615799

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