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Properties

Label 2-1386-7.4-c1-0-11
Degree $2$
Conductor $1386$
Sign $-0.496 - 0.868i$
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 + (2.20 + 3.82i)5-s + (2.20 − 1.45i)7-s − 0.999·8-s + (−2.20 + 3.82i)10-s + (−0.5 + 0.866i)11-s + (2.36 + 1.18i)14-s + (−0.5 − 0.866i)16-s + (1.18 − 2.05i)17-s + (1.84 + 3.19i)19-s − 4.41·20-s − 0.999·22-s + (2.36 + 4.09i)23-s + (−7.26 + 12.5i)25-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.988 + 1.71i)5-s + (0.835 − 0.550i)7-s − 0.353·8-s + (−0.698 + 1.21i)10-s + (−0.150 + 0.261i)11-s + (0.632 + 0.316i)14-s + (−0.125 − 0.216i)16-s + (0.288 − 0.499i)17-s + (0.423 + 0.732i)19-s − 0.988·20-s − 0.213·22-s + (0.493 + 0.854i)23-s + (−1.45 + 2.51i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.496 - 0.868i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.496 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.538662097\)
\(L(\frac12)\) \(\approx\) \(2.538662097\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 1.45i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-2.20 - 3.82i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-1.18 + 2.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.84 - 3.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.52T + 29T^{2} \)
31 \( 1 + (-1.68 + 2.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.84 + 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 3.06T + 43T^{2} \)
47 \( 1 + (2.05 + 3.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.26 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.89 + 3.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.07 - 8.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.934T + 71T^{2} \)
73 \( 1 + (-3.73 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.209 - 0.362i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.14T + 83T^{2} \)
89 \( 1 + (6.41 + 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.10957044550294294882108938574, −9.045545821968657391557023809926, −7.86334791723566610481374123393, −7.26288706460141381494115638464, −6.68316616576419557296261093691, −5.73922383070219067175321820446, −5.05976321585294956433358599872, −3.76374392533741773939779140350, −2.90148668708684642037799697480, −1.74764750481368439477182597497, 0.998266759730272012902232994852, 1.82515632406501849460644153679, 2.94972481159395889462873051883, 4.58234238681438709194999843555, 4.90920284068658661807126604591, 5.65260812489197469379042797558, 6.54820625329562049554910046232, 8.199453231366025966081431009602, 8.575269113223018827027949979462, 9.275625573554267569563334043416

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