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Properties

Label 2-1386-21.5-c1-0-7
Degree $2$
Conductor $1386$
Sign $-0.0455 - 0.998i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (−0.651 + 1.12i)5-s + (−0.212 − 2.63i)7-s + 0.999i·8-s + (−1.12 + 0.651i)10-s + (−0.866 + 0.5i)11-s + 1.37i·13-s + (1.13 − 2.39i)14-s + (−0.5 + 0.866i)16-s + (2.86 + 4.95i)17-s + (−0.481 − 0.277i)19-s − 1.30·20-s − 0.999·22-s + (5.02 + 2.89i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.291 + 0.504i)5-s + (−0.0803 − 0.996i)7-s + 0.353i·8-s + (−0.356 + 0.205i)10-s + (−0.261 + 0.150i)11-s + 0.380i·13-s + (0.303 − 0.638i)14-s + (−0.125 + 0.216i)16-s + (0.694 + 1.20i)17-s + (−0.110 − 0.0637i)19-s − 0.291·20-s − 0.213·22-s + (1.04 + 0.604i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0455 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0455 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0455 - 0.998i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.0455 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.065964861\)
\(L(\frac12)\) \(\approx\) \(2.065964861\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.212 + 2.63i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.651 - 1.12i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.37iT - 13T^{2} \)
17 \( 1 + (-2.86 - 4.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.481 + 0.277i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.02 - 2.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.53iT - 29T^{2} \)
31 \( 1 + (-0.660 + 0.381i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.34 - 5.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.25T + 41T^{2} \)
43 \( 1 - 3.91T + 43T^{2} \)
47 \( 1 + (0.483 - 0.837i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.40 + 4.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.499 - 0.865i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.127 - 0.0735i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.60 + 6.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + (1.44 - 0.831i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.361 + 0.626i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.54T + 83T^{2} \)
89 \( 1 + (4.25 - 7.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−9.936716376593245347026397624990, −8.827879409767613713940188402343, −7.934023742449608414840516224960, −7.14475038933192419765928388448, −6.71346931984537292145890995904, −5.59531955997930753822031396889, −4.70154623310155554812366347238, −3.71571813117642957177362354374, −3.10325011278767706748945349109, −1.47647268443988747230452446097, 0.72331133547626900941021669714, 2.36763070872023002161232541560, 3.09500344372226514293813936199, 4.32247270602330316971633832936, 5.18267807321703452295416559894, 5.75280301342437002286093933904, 6.80572344065341969685823946170, 7.79542644379745817429145300650, 8.667560801820678878268435034165, 9.345154715515240781484411465374

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