Properties

Label 2-1386-7.4-c1-0-1
Degree $2$
Conductor $1386$
Sign $0.386 - 0.922i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + 2·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + (3 + 5.19i)19-s − 0.999·20-s − 0.999·22-s + (−3.5 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + 0.554·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + (0.688 + 1.19i)19-s − 0.223·20-s − 0.213·22-s + (−0.729 − 1.26i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8149416646\)
\(L(\frac12)\) \(\approx\) \(0.8149416646\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + (-6 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−9.851439868380208394489644934443, −9.021264943123153085011227062122, −8.302214666357111984571158427533, −7.35418970922135567860569920951, −6.37979775787264892846013391132, −5.83285281244924969927521519655, −4.26194123883535688924601137073, −3.58219645814744840782560104815, −2.61276875405228630328051999727, −1.32251688516892974383587393045, 0.39173942278724470151872436545, 2.01563993210522905304771047129, 3.33501319882675610036449611930, 4.44020669765414236443630727933, 5.58151589743541318394326982153, 5.99174435709958435634642022734, 7.24135324232164606746908589139, 7.49938594597107300060627663411, 8.955934164237614169862937553666, 9.329089323766612776114813287527

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