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Properties

Label 2-1386-21.5-c1-0-1
Degree $2$
Conductor $1386$
Sign $-0.827 + 0.561i$
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 + (−1.36 + 2.35i)5-s + (−2.13 + 1.56i)7-s + 0.999i·8-s + (−2.35 + 1.36i)10-s + (−0.866 + 0.5i)11-s + 0.193i·13-s + (−2.63 + 0.284i)14-s + (−0.5 + 0.866i)16-s + (−3.07 − 5.32i)17-s + (0.269 + 0.155i)19-s − 2.72·20-s − 0.999·22-s + (−4.50 − 2.60i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.608 + 1.05i)5-s + (−0.807 + 0.590i)7-s + 0.353i·8-s + (−0.745 + 0.430i)10-s + (−0.261 + 0.150i)11-s + 0.0537i·13-s + (−0.703 + 0.0759i)14-s + (−0.125 + 0.216i)16-s + (−0.745 − 1.29i)17-s + (0.0617 + 0.0356i)19-s − 0.608·20-s − 0.213·22-s + (−0.939 − 0.542i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.827 + 0.561i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6317507105\)
\(L(\frac12)\) \(\approx\) \(0.6317507105\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 1.56i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (1.36 - 2.35i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.193iT - 13T^{2} \)
17 \( 1 + (3.07 + 5.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.269 - 0.155i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.50 + 2.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.17iT - 29T^{2} \)
31 \( 1 + (-1.38 + 0.797i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.21 + 3.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.29T + 41T^{2} \)
43 \( 1 - 4.10T + 43T^{2} \)
47 \( 1 + (3.63 - 6.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.04 + 0.603i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.43 + 9.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.20 + 4.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.69 - 4.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.63iT - 71T^{2} \)
73 \( 1 + (2.50 - 1.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.60 - 13.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (5.67 - 9.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.77iT - 97T^{2} \)
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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

−10.05131537964948449985675201599, −9.249359422838197744193151230136, −8.279956169285980833457054981767, −7.34984720106163030757166251846, −6.78574374096112113942449904559, −6.07839319238409056282934583499, −5.05702917448791292624993896365, −4.03277339110081767932653782875, −3.06765211680320165845946986837, −2.43426753300959570412367656448, 0.19796605277603088864165230124, 1.62056941701606854712111139724, 3.08251098264913935514192409701, 4.07906891529301491850942138381, 4.52580408452991927588135770396, 5.74240183043992178982906640655, 6.41636031262311802373473533405, 7.51770995546945237317255674392, 8.305877115021244615199453998690, 9.108489905385948599597990799819

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