Properties

Degree $2$
Conductor $1386$
Sign $0.259 + 0.965i$
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 + (1.78 + 1.02i)5-s + (−0.289 + 2.62i)7-s + 0.999·8-s + (−1.78 + 1.02i)10-s + (−3.29 − 0.406i)11-s − 6.97i·13-s + (−2.13 − 1.56i)14-s + (−0.5 + 0.866i)16-s + (−2.66 − 4.61i)17-s + (−7.36 − 4.25i)19-s − 2.05i·20-s + (1.99 − 2.64i)22-s + (1.35 + 0.782i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.797 + 0.460i)5-s + (−0.109 + 0.993i)7-s + 0.353·8-s + (−0.563 + 0.325i)10-s + (−0.992 − 0.122i)11-s − 1.93i·13-s + (−0.569 − 0.418i)14-s + (−0.125 + 0.216i)16-s + (−0.646 − 1.11i)17-s + (−1.69 − 0.975i)19-s − 0.460i·20-s + (0.425 − 0.564i)22-s + (0.282 + 0.163i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.259 + 0.965i$
Motivic weight: \(1\)
Character: $\chi_{1386} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6630169150\)
\(L(\frac12)\) \(\approx\) \(0.6630169150\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.289 - 2.62i)T \)
11 \( 1 + (3.29 + 0.406i)T \)
good5 \( 1 + (-1.78 - 1.02i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 6.97iT - 13T^{2} \)
17 \( 1 + (2.66 + 4.61i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.36 + 4.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.35 - 0.782i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.87T + 29T^{2} \)
31 \( 1 + (4.39 + 7.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.42T + 41T^{2} \)
43 \( 1 - 7.55iT - 43T^{2} \)
47 \( 1 + (-10.2 - 5.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.93 + 5.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.49 - 2.01i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.39 - 2.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.75 + 8.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.30iT - 71T^{2} \)
73 \( 1 + (-0.828 + 0.478i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.75 + 4.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.50T + 83T^{2} \)
89 \( 1 + (2.99 + 1.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.95T + 97T^{2} \)
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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

−9.310515212810339015284529019537, −8.579767132607564149686463593539, −7.86872517407382578110309622399, −6.92385599261376954582477460458, −6.00872030375361280313232911419, −5.52936685509802983002300463703, −4.65209402058216871205448280197, −2.82405045002716253084987122773, −2.38221038268867720384691342040, −0.28237645536503717670614222059, 1.58731698151219985990120535645, 2.20323592946772645542370722658, 3.86519194938091777661429774558, 4.37153181106695402647255482254, 5.53631215229131402359259044388, 6.62227705199378035450060803665, 7.26421043388769735871293080774, 8.588024252782134954856190837086, 8.810802008355306475631688356492, 9.907687877795823435430284931071

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