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Properties

Label 2-1386-33.2-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.799 - 0.600i$
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 + (3.13 − 1.01i)5-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s + 3.29i·10-s + (3.17 + 0.943i)11-s + (6.67 + 2.17i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.210 − 0.649i)17-s + (−3.84 − 5.29i)19-s + (−3.13 − 1.01i)20-s + (−1.88 + 2.73i)22-s + 1.86i·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (1.40 − 0.455i)5-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s + 1.04i·10-s + (0.958 + 0.284i)11-s + (1.85 + 0.601i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.0511 − 0.157i)17-s + (−0.882 − 1.21i)19-s + (−0.700 − 0.227i)20-s + (−0.400 + 0.582i)22-s + 0.388i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.799 - 0.600i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041570031\)
\(L(\frac12)\) \(\approx\) \(2.041570031\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.17 - 0.943i)T \)
good5 \( 1 + (-3.13 + 1.01i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-6.67 - 2.17i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.210 + 0.649i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.84 + 5.29i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.86iT - 23T^{2} \)
29 \( 1 + (5.86 + 4.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.831 + 2.55i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.03 - 5.11i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.41 - 6.11i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.19iT - 43T^{2} \)
47 \( 1 + (-1.15 - 1.58i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-10.4 - 3.38i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.48 - 8.93i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.89 + 3.21i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.29T + 67T^{2} \)
71 \( 1 + (-0.0882 + 0.0286i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.91 + 9.51i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-9.12 - 2.96i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.68 + 11.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.71iT - 89T^{2} \)
97 \( 1 + (3.73 - 11.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.250204686524625755621890976488, −9.092408993886306418113693567308, −8.272406420246034968082820460986, −6.94692311789874477705805370785, −6.25011890555875108285781657230, −5.86569367653982560271550868759, −4.75461243148218120592498871827, −3.81439511186069552696224391254, −2.19689041817537726512331535094, −1.19171811842771764126905019207, 1.20431542997766129835813803635, 2.06403981321598006543339939747, 3.39267245802130941892546168213, 3.97707220919455729864547315215, 5.55241103674278566119559092657, 6.14040718924035202394997954860, 6.84194000807009098433617043754, 8.215459333279185660773220292755, 8.833008879655555190909156815346, 9.606875340124242374963827934457

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