Properties

Label 2-1386-1.1-c1-0-15
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 11-s + 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 2·20-s − 22-s + 4·23-s − 25-s + 2·26-s + 28-s − 2·29-s − 4·31-s + 32-s + 6·34-s + 2·35-s − 2·37-s − 4·38-s + 2·40-s + 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.447·20-s − 0.213·22-s + 0.834·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.328·37-s − 0.648·38-s + 0.316·40-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.291260857\)
\(L(\frac12)\) \(\approx\) \(3.291260857\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−9.624008951945022350971422712414, −8.817417727367457424810293491861, −7.82491759382859411774440769361, −7.06103937421558916977777144566, −5.88974476808873252053152463118, −5.62472658216482534683012385296, −4.54468764173297491130716278035, −3.52268894932194979183855581758, −2.43951823946059783270435316404, −1.37340800303348386328024038090, 1.37340800303348386328024038090, 2.43951823946059783270435316404, 3.52268894932194979183855581758, 4.54468764173297491130716278035, 5.62472658216482534683012385296, 5.88974476808873252053152463118, 7.06103937421558916977777144566, 7.82491759382859411774440769361, 8.817417727367457424810293491861, 9.624008951945022350971422712414

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