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Properties

Label 2-1386-33.2-c1-0-23
Degree $2$
Conductor $1386$
Sign $-0.114 + 0.993i$
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 + (1.81 − 0.591i)5-s + (0.587 − 0.809i)7-s + (0.809 − 0.587i)8-s + 1.91i·10-s + (−2.51 − 2.15i)11-s + (−2.40 − 0.781i)13-s + (0.587 + 0.809i)14-s + (0.309 + 0.951i)16-s + (−1.61 − 4.96i)17-s + (−1.99 − 2.73i)19-s + (−1.81 − 0.591i)20-s + (2.82 − 1.72i)22-s + 0.978i·23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (0.813 − 0.264i)5-s + (0.222 − 0.305i)7-s + (0.286 − 0.207i)8-s + 0.604i·10-s + (−0.759 − 0.650i)11-s + (−0.666 − 0.216i)13-s + (0.157 + 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.391 − 1.20i)17-s + (−0.456 − 0.628i)19-s + (−0.406 − 0.132i)20-s + (0.603 − 0.368i)22-s + 0.204i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8239131394\)
\(L(\frac12)\) \(\approx\) \(0.8239131394\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 + 2.15i)T \)
good5 \( 1 + (-1.81 + 0.591i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.40 + 0.781i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.61 + 4.96i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.99 + 2.73i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.978iT - 23T^{2} \)
29 \( 1 + (6.64 + 4.82i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.58 - 4.87i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.15 - 2.29i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.37 - 2.45i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.26iT - 43T^{2} \)
47 \( 1 + (2.86 + 3.94i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (11.8 + 3.84i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.71 - 3.73i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.970 + 0.315i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 1.31T + 67T^{2} \)
71 \( 1 + (-4.69 + 1.52i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.95 + 4.06i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-12.9 - 4.21i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.38 + 4.26i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 5.41iT - 89T^{2} \)
97 \( 1 + (-0.223 + 0.686i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.463169992000757110649146078808, −8.488908802614059779624553591472, −7.72485525626083228060625368316, −6.97065055090794431797650038878, −6.03549575614694851502760515963, −5.22414412596759854821497095190, −4.64206840119267084311192964808, −3.14288580196882756367705928554, −1.94413085736338451477738847126, −0.33007728490339801045757223240, 1.87083137803417628514849833682, 2.28259077144466637128347671590, 3.64448041406154006860807314233, 4.68038446055521690408731680177, 5.59370707862548694940442937500, 6.43722244208925027052693845915, 7.56307645792446683391424560143, 8.236990207131478083886283046696, 9.275535376140553259377666089699, 9.780526022777262282216687872865

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