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Properties

Label 2-1386-33.2-c1-0-1
Degree $2$
Conductor $1386$
Sign $0.231 - 0.972i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.599 − 0.194i)5-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s − 0.629i·10-s + (−3.04 + 1.30i)11-s + (−2.38 − 0.775i)13-s + (0.587 + 0.809i)14-s + (0.309 + 0.951i)16-s + (0.701 + 2.15i)17-s + (0.0702 + 0.0966i)19-s + (−0.599 − 0.194i)20-s + (0.297 + 3.30i)22-s + 6.23i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.267 − 0.0870i)5-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s − 0.199i·10-s + (−0.919 + 0.393i)11-s + (−0.661 − 0.215i)13-s + (0.157 + 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.170 + 0.523i)17-s + (0.0161 + 0.0221i)19-s + (−0.133 − 0.0435i)20-s + (0.0633 + 0.704i)22-s + 1.29i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8134370789\)
\(L(\frac12)\) \(\approx\) \(0.8134370789\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (3.04 - 1.30i)T \)
good5 \( 1 + (-0.599 + 0.194i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.38 + 0.775i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.701 - 2.15i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.0702 - 0.0966i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 + (-1.61 - 1.17i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.811 + 2.49i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0380 + 0.0276i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.70 - 4.14i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.49iT - 43T^{2} \)
47 \( 1 + (-4.12 - 5.68i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.3 + 3.35i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.91 - 10.8i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.30 - 1.07i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.558T + 67T^{2} \)
71 \( 1 + (0.452 - 0.147i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.88 + 12.2i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-8.26 - 2.68i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.75 - 5.41i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (0.681 - 2.09i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.726008312722521291584575325949, −9.307594759299428586394796831414, −8.099545527885049800028399103154, −7.49809030789781887645415391526, −6.24425437302437188482807942010, −5.43599596492065125199788895743, −4.72922052387544710363146838161, −3.53551282146848129873103117813, −2.62196253885569867597984729423, −1.58335318784971662075363425561, 0.29278051888413835931765879705, 2.30537248365212894694604660872, 3.32181103048647614055159763884, 4.52792225685700475646054872259, 5.21827564426843427743225850443, 6.18609023201660925283676592308, 6.89110209884498168264763929166, 7.75109405455589332898571046786, 8.417857914596659994990563767319, 9.369173073085022386748248644897

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