Properties

Degree $2$
Conductor $1386$
Sign $0.988 - 0.151i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.03 + 0.598i)5-s + (2.52 − 0.782i)7-s + 0.999·8-s + (−1.03 + 0.598i)10-s + (−2.65 − 1.98i)11-s − 3.26i·13-s + (−0.586 + 2.57i)14-s + (−0.5 + 0.866i)16-s + (2.21 + 3.82i)17-s + (6.73 + 3.88i)19-s − 1.19i·20-s + (3.04 − 1.30i)22-s + (−5.94 − 3.43i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.463 + 0.267i)5-s + (0.955 − 0.295i)7-s + 0.353·8-s + (−0.327 + 0.189i)10-s + (−0.800 − 0.599i)11-s − 0.904i·13-s + (−0.156 + 0.689i)14-s + (−0.125 + 0.216i)16-s + (0.536 + 0.928i)17-s + (1.54 + 0.892i)19-s − 0.267i·20-s + (0.650 − 0.277i)22-s + (−1.24 − 0.715i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.988 - 0.151i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613373526\)
\(L(\frac12)\) \(\approx\) \(1.613373526\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.52 + 0.782i)T \)
11 \( 1 + (2.65 + 1.98i)T \)
good5 \( 1 + (-1.03 - 0.598i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.26iT - 13T^{2} \)
17 \( 1 + (-2.21 - 3.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.73 - 3.88i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.94 + 3.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.85T + 29T^{2} \)
31 \( 1 + (0.103 + 0.179i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.83 + 6.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.15T + 41T^{2} \)
43 \( 1 + 8.00iT - 43T^{2} \)
47 \( 1 + (-0.355 - 0.205i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.02 - 2.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.66 - 2.11i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.94 + 5.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.61 - 9.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + (-0.195 + 0.113i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.00 - 1.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-11.4 - 6.58i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−9.696715616356575896266733216163, −8.436212554339172280666149260481, −7.996216375461652034084185149980, −7.45346538247448971061399270033, −6.05023101592860796940240908804, −5.73920122325092693360431602781, −4.74539564978041676895471094877, −3.55939786777675373112761280324, −2.25167460977169578300517336908, −0.889117499584959341963286712742, 1.20940014351285856381503689696, 2.19825174328447752609508069237, 3.18848777385339971980191450789, 4.73448060445760164325093326328, 4.99692409042463038113774899437, 6.21506858685592424935986710691, 7.58281836256804092440792452740, 7.80118404558286369185399387349, 9.034730552111588945404275698793, 9.557981189816515221434497759743

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