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Properties

Label 2-1386-21.17-c1-0-3
Degree $2$
Conductor $1386$
Sign $-0.926 - 0.375i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.926 - 0.375i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8186933243\)
\(L(\frac12)\) \(\approx\) \(0.8186933243\)
\(L(\frac{3}{2})\) 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

−9.856018465688046695285665753586, −9.010155540858109885099543890115, −8.447704201423699398775948656265, −7.57734518253536427037365894658, −6.55552120987666947331693511378, −6.04444195587145458219407445468, −5.19014937020760606201413613216, −3.87394594259170462794915725137, −2.62216766567017043551143617664, −1.69435412523915166197143901741, 0.39908220100138485356697821841, 1.63512435834203550198892371113, 2.85689034276236511408736248254, 4.08907592673324234148155426398, 4.76595277993552219205750023841, 6.10695552584008196724588794806, 6.90739289902966622096093786538, 7.68920703095296992672145296273, 8.542619214753657169943459980023, 9.358491663591190782059113113272

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