Properties

Degree $2$
Conductor $1386$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.09·5-s + 7-s + 8-s + 1.09·10-s + 11-s + 6.80·13-s + 14-s + 16-s + 1.09·17-s − 3.89·19-s + 1.09·20-s + 22-s − 6.99·23-s − 3.80·25-s + 6.80·26-s + 28-s + 4.80·29-s − 1.27·31-s + 32-s + 1.09·34-s + 1.09·35-s + 4.18·37-s − 3.89·38-s + 1.09·40-s − 1.09·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.488·5-s + 0.377·7-s + 0.353·8-s + 0.345·10-s + 0.301·11-s + 1.88·13-s + 0.267·14-s + 0.250·16-s + 0.264·17-s − 0.894·19-s + 0.244·20-s + 0.213·22-s − 1.45·23-s − 0.761·25-s + 1.33·26-s + 0.188·28-s + 0.892·29-s − 0.229·31-s + 0.176·32-s + 0.187·34-s + 0.184·35-s + 0.687·37-s − 0.632·38-s + 0.172·40-s − 0.170·41-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1386} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.194887564\)
\(L(\frac12)\) \(\approx\) \(3.194887564\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.09T + 5T^{2} \)
13 \( 1 - 6.80T + 13T^{2} \)
17 \( 1 - 1.09T + 17T^{2} \)
19 \( 1 + 3.89T + 19T^{2} \)
23 \( 1 + 6.99T + 23T^{2} \)
29 \( 1 - 4.80T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 4.18T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 9.71T + 47T^{2} \)
53 \( 1 - 3.81T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 8.99T + 67T^{2} \)
71 \( 1 + 9.17T + 71T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 + 4.99T + 79T^{2} \)
83 \( 1 + 7.09T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 7.61T + 97T^{2} \)
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

−9.717979508120248306757582292885, −8.552777985617149827046325058896, −8.111583538292374115460263718703, −6.86783298906984883531605130672, −6.06133985051057111849455336419, −5.62572784863285190930442472674, −4.28350688314488872518318196562, −3.75698826294922420259469282777, −2.39503063058201151847913837817, −1.35060704710633302988938480794, 1.35060704710633302988938480794, 2.39503063058201151847913837817, 3.75698826294922420259469282777, 4.28350688314488872518318196562, 5.62572784863285190930442472674, 6.06133985051057111849455336419, 6.86783298906984883531605130672, 8.111583538292374115460263718703, 8.552777985617149827046325058896, 9.717979508120248306757582292885

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