Properties

Label 2-1386-7.2-c1-0-16
Degree $2$
Conductor $1386$
Sign $0.386 + 0.922i$
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 + (−0.5 + 0.866i)5-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 2·13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + (3 − 5.19i)19-s + 0.999·20-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 + (−0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (0.158 + 0.273i)10-s + (−0.150 − 0.261i)11-s + 0.554·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + (0.688 − 1.19i)19-s + 0.223·20-s − 0.213·22-s + (0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.386 + 0.922i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665327604\)
\(L(\frac12)\) \(\approx\) \(1.665327604\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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

−9.384282573028651789534262960905, −8.912745131488240073196171624940, −7.83387624338365706137608638082, −6.78158456778465121390164540720, −6.10146751378147095158105901835, −5.21766004599126879123267303568, −4.05529041732218200274643882425, −3.21729307601242294853268920414, −2.46910547395423037472204259919, −0.77161620851336411024148920747, 1.07291782668336743015093228662, 3.03836291788202732639667413891, 3.64370442573907985162946833372, 4.84087452066317297379660552526, 5.50847474699024380613983410841, 6.55421423571550900349679512308, 7.16336257771782792552786202175, 8.026861769763379612240257243805, 8.803437196083605841855088858414, 9.722942406712210365253753669600

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