Properties

Degree $2$
Conductor $1386$
Sign $-0.787 + 0.615i$
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 + (−2.42 − 1.39i)5-s + (2.62 + 0.299i)7-s + 0.999·8-s + (2.42 − 1.39i)10-s + (−1.93 − 2.69i)11-s − 3.20i·13-s + (−1.57 + 2.12i)14-s + (−0.5 + 0.866i)16-s + (2.05 + 3.56i)17-s + (−1.82 − 1.05i)19-s + 2.79i·20-s + (3.30 − 0.323i)22-s + (−0.902 − 0.521i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.08 − 0.625i)5-s + (0.993 + 0.113i)7-s + 0.353·8-s + (0.765 − 0.442i)10-s + (−0.582 − 0.813i)11-s − 0.888i·13-s + (−0.420 + 0.568i)14-s + (−0.125 + 0.216i)16-s + (0.499 + 0.864i)17-s + (−0.419 − 0.242i)19-s + 0.625i·20-s + (0.703 − 0.0690i)22-s + (−0.188 − 0.108i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.787 + 0.615i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.787 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2949826208\)
\(L(\frac12)\) \(\approx\) \(0.2949826208\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.299i)T \)
11 \( 1 + (1.93 + 2.69i)T \)
good5 \( 1 + (2.42 + 1.39i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.82 + 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.902 + 0.521i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.40T + 29T^{2} \)
31 \( 1 + (-3.58 - 6.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.60 - 7.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.110T + 41T^{2} \)
43 \( 1 + 4.20iT - 43T^{2} \)
47 \( 1 + (7.27 + 4.19i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 + 5.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.34 - 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.21 + 5.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.97 + 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-5.46 + 3.15i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.4 + 7.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (10.3 + 5.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−8.767995613847483148797319119263, −8.312452099281140417365178702212, −7.946972279460331654178882742877, −7.05346492749994134829555098500, −5.78514181011956440501770362848, −5.18759538838951168760828655827, −4.28549104947783055333406007699, −3.24396830842428786717009942674, −1.51512181152388064364847703259, −0.13878783843212636325890827024, 1.68748051499160780784609735532, 2.71494290945142582274297849271, 3.95245799038674213906871590589, 4.47660503782365970808832204453, 5.61951301492266484764206798487, 7.14124871150676565488634143988, 7.50431870095227787242615403244, 8.165712799514156060486308180959, 9.188931646232708413661290155213, 9.930301584259639277708052763921

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