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Properties

Label 2-1386-33.2-c1-0-20
Degree $2$
Conductor $1386$
Sign $-0.908 - 0.418i$
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 + (−2.17 + 0.708i)5-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s + 2.29i·10-s + (−0.675 − 3.24i)11-s + (3.85 + 1.25i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (−0.654 − 2.01i)17-s + (−2.04 − 2.81i)19-s + (2.17 + 0.708i)20-s + (−3.29 − 0.360i)22-s + 4.13i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.974 + 0.316i)5-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s + 0.724i·10-s + (−0.203 − 0.979i)11-s + (1.06 + 0.347i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.158 − 0.488i)17-s + (−0.469 − 0.646i)19-s + (0.487 + 0.158i)20-s + (−0.702 − 0.0768i)22-s + 0.863i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.908 - 0.418i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4411088305\)
\(L(\frac12)\) \(\approx\) \(0.4411088305\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.675 + 3.24i)T \)
good5 \( 1 + (2.17 - 0.708i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-3.85 - 1.25i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.654 + 2.01i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.04 + 2.81i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.13iT - 23T^{2} \)
29 \( 1 + (5.33 + 3.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.408 - 1.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.69 + 4.86i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.63 - 6.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.09iT - 43T^{2} \)
47 \( 1 + (4.08 + 5.62i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.58 + 2.79i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.63 - 2.24i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.23 + 1.37i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 + (9.93 - 3.22i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.47 + 7.53i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (14.1 + 4.60i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.27 - 7.01i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (3.69 - 11.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.079315147906520700491436437216, −8.395943183858912863669429583993, −7.60655632650439271133949241220, −6.65945926076187460812042878576, −5.68292772187751480303647249590, −4.64623735363658679130170792743, −3.69617107972547140754767841798, −3.17896804660007076623499804357, −1.66517818148869676928864847525, −0.16658405880850547490531269245, 1.76549312440923978195959618597, 3.40408082500060691313359409710, 4.16861633668243540824522499018, 4.99048943962091926119320010383, 5.92274829818454947868275065647, 6.83047670189234585210372812047, 7.65313082519100737392034478763, 8.415577468482613002771367718493, 8.784986051579408019347643823063, 10.01952794545658270219330466287

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