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Properties

Label 2-1386-21.5-c1-0-20
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: $\chi_{1386} (89, \cdot )$
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

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

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