Properties

Degree $2$
Conductor $1386$
Sign $0.999 - 0.0226i$
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 + (0.227 + 0.393i)5-s + (−0.227 − 2.63i)7-s − 0.999·8-s + (−0.227 + 0.393i)10-s + (0.5 − 0.866i)11-s + 0.909·13-s + (2.16 − 1.51i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−1.94 − 3.36i)19-s − 0.454·20-s + 0.999·22-s + (4.16 + 7.22i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.101 + 0.176i)5-s + (−0.0859 − 0.996i)7-s − 0.353·8-s + (−0.0719 + 0.124i)10-s + (0.150 − 0.261i)11-s + 0.252·13-s + (0.579 − 0.404i)14-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + (−0.445 − 0.771i)19-s − 0.101·20-s + 0.213·22-s + (0.869 + 1.50i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0226i$
Motivic weight: \(1\)
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.999 - 0.0226i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.938157522\)
\(L(\frac12)\) \(\approx\) \(1.938157522\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.227 + 2.63i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.227 - 0.393i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.909T + 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.94 + 3.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.16 - 7.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.79T + 29T^{2} \)
31 \( 1 + (-4.79 + 8.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.94 + 6.82i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 - 7.88T + 43T^{2} \)
47 \( 1 + (-0.169 - 0.292i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.454 + 0.787i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.48 + 6.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.02 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.89 + 3.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.79T + 71T^{2} \)
73 \( 1 + (5.33 - 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.68 + 4.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.97T + 83T^{2} \)
89 \( 1 + (5.54 + 9.60i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−9.479101817037778419385868223014, −8.702878713133249219418507037858, −7.74112982497954375156088542897, −7.10488152462583994994607882377, −6.38015057413996319409498732364, −5.47726888508293089062944788536, −4.46059525284850912912553513194, −3.72369387986976542221537858866, −2.61022382274300642236050991269, −0.811291647395531261001553647214, 1.27280395697751210010120275782, 2.46337452148674481529389724608, 3.32835945909119646539991780145, 4.53453936305293463289359683384, 5.23727041209450013897514493113, 6.18538100385252059125607577411, 6.90100745126282702107634942755, 8.373098125662518173153687029966, 8.718753309095067960600363320794, 9.649044413000977834842627132867

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