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Properties

Label 2-1386-33.2-c1-0-6
Degree $2$
Conductor $1386$
Sign $-0.132 - 0.991i$
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 + (1.69 − 0.550i)5-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s + 1.78i·10-s + (−2.97 − 1.47i)11-s + (0.352 + 0.114i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (1.79 + 5.51i)17-s + (−0.665 − 0.915i)19-s + (−1.69 − 0.550i)20-s + (2.32 − 2.36i)22-s + 5.35i·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.758 − 0.246i)5-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s + 0.563i·10-s + (−0.895 − 0.444i)11-s + (0.0977 + 0.0317i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.434 + 1.33i)17-s + (−0.152 − 0.210i)19-s + (−0.379 − 0.123i)20-s + (0.494 − 0.505i)22-s + 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.132 - 0.991i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.132 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.374137844\)
\(L(\frac12)\) \(\approx\) \(1.374137844\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.97 + 1.47i)T \)
good5 \( 1 + (-1.69 + 0.550i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.352 - 0.114i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.79 - 5.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.665 + 0.915i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.35iT - 23T^{2} \)
29 \( 1 + (-6.33 - 4.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.14 - 5.19i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.50 - 3.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.57iT - 43T^{2} \)
47 \( 1 + (-7.43 - 10.2i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-11.8 - 3.83i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.16 + 11.2i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.20 - 2.33i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.02T + 67T^{2} \)
71 \( 1 + (7.04 - 2.28i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.87 - 9.45i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.62 - 1.82i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.46 - 7.58i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.106iT - 89T^{2} \)
97 \( 1 + (-5.52 + 17.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.776549919989551086497176045966, −8.848897051345580333094142039196, −8.250724823004304392803493465647, −7.43323641906696635984677736854, −6.34115294807414432287652775197, −5.75255678894090746911422037442, −5.10669198195138462721838072596, −3.86239917225583557868784986345, −2.62370949679554267246210798778, −1.28158808177416406865847883989, 0.65591515586041575023811153604, 2.26324122735359040424892004812, 2.83531774071002505442168140784, 4.16645960775843433082127087877, 5.05753708582527458133473039349, 6.01074185347616098053421801770, 6.99727585164758169575529288361, 7.80846966117486167103603809466, 8.711995013843864343716482485747, 9.599529530659489277470007765995

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