Properties

Label 2-1386-7.2-c1-0-8
Degree $2$
Conductor $1386$
Sign $0.605 - 0.795i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + (2 + 1.73i)7-s − 0.999·8-s + (0.999 + 1.73i)10-s + (−0.5 − 0.866i)11-s + 2·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + (−3.5 + 6.06i)19-s + 1.99·20-s − 0.999·22-s + (−3.5 + 6.06i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.755 + 0.654i)7-s − 0.353·8-s + (0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + 0.554·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + (−0.802 + 1.39i)19-s + 0.447·20-s − 0.213·22-s + (−0.729 + 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.571676620\)
\(L(\frac12)\) \(\approx\) \(1.571676620\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15T + 97T^{2} \)
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

−9.895705721444759404942649470511, −8.864229087123546453339807153033, −8.162153498355008181083053194122, −7.33121403197587218182134267617, −6.18558141707468322921965351351, −5.54443811769757547149259025990, −4.43643586564463960772723892275, −3.57117051866448721690388977488, −2.62413673213178740729403676099, −1.52641916201463872689615329654, 0.58397585232842752378855491801, 2.18846439817452727018719664461, 3.76282872200635909759487602606, 4.55422991061390050910049718472, 4.97362124359990972677763714903, 6.30391235869713457883867170346, 6.91663142810385347500617003011, 8.021769692244951355312203246700, 8.410169031621487197841065526369, 9.103365845978011882865713404101

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