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Properties

Label 2-1386-7.4-c1-0-26
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

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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: $\chi_{1386} (991, \cdot )$
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 \( 1 - 15T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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