Properties

Degree $2$
Conductor $1386$
Sign $0.589 - 0.807i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.741 − 1.28i)5-s + (−1.48 + 2.19i)7-s + 0.999i·8-s + (1.28 − 0.741i)10-s + (0.866 − 0.5i)11-s − 2.44i·13-s + (−2.38 + 1.15i)14-s + (−0.5 + 0.866i)16-s + (−0.182 − 0.315i)17-s + (7.06 + 4.07i)19-s + 1.48·20-s + 0.999·22-s + (5.18 + 2.99i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.331 − 0.574i)5-s + (−0.560 + 0.827i)7-s + 0.353i·8-s + (0.405 − 0.234i)10-s + (0.261 − 0.150i)11-s − 0.679i·13-s + (−0.636 + 0.308i)14-s + (−0.125 + 0.216i)16-s + (−0.0441 − 0.0765i)17-s + (1.62 + 0.935i)19-s + 0.331·20-s + 0.213·22-s + (1.08 + 0.623i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.589 - 0.807i$
Motivic weight: \(1\)
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.589 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.502934616\)
\(L(\frac12)\) \(\approx\) \(2.502934616\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (1.48 - 2.19i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-0.741 + 1.28i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (0.182 + 0.315i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.06 - 4.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.18 - 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.332iT - 29T^{2} \)
31 \( 1 + (-0.752 + 0.434i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.64 - 8.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.38T + 41T^{2} \)
43 \( 1 + 2.94T + 43T^{2} \)
47 \( 1 + (0.637 - 1.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.65 + 2.68i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.28 + 4.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.08 + 8.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.07iT - 71T^{2} \)
73 \( 1 + (7.86 - 4.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 + 5.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + (-0.459 + 0.796i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.79iT - 97T^{2} \)
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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

−9.487119762613400658446757855286, −8.967622198923547630215429433378, −8.011453985439188058499446645674, −7.20019546747157297170420270925, −6.17657850083899196323243757083, −5.48494344855047988125820451381, −4.95339695684235041875694511420, −3.53231458111168320101523058175, −2.87661057309070680041859703242, −1.33662354323557497185627644563, 0.966540208665610387484184556035, 2.46362859258588318191638001736, 3.30645694263501414407191431435, 4.25408156918623769616882735816, 5.14346264167712771208547792527, 6.23732152754843106814116191147, 6.94581610870495287229324351801, 7.43494636177320385006816349995, 8.969722394836420029575828386081, 9.536616335669846694486338120902

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