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Properties

Label 2-1386-33.8-c1-0-1
Degree $2$
Conductor $1386$
Sign $-0.0594 - 0.998i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.943 + 1.29i)5-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + 1.60i·10-s + (−1.01 + 3.15i)11-s + (−1.22 − 1.68i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (4.11 + 2.98i)17-s + (−6.93 + 2.25i)19-s + (0.943 + 1.29i)20-s + (1.03 + 3.15i)22-s − 0.571i·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.422 + 0.580i)5-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + 0.507i·10-s + (−0.306 + 0.951i)11-s + (−0.338 − 0.466i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.997 + 0.724i)17-s + (−1.59 + 0.516i)19-s + (0.211 + 0.290i)20-s + (0.220 + 0.671i)22-s − 0.119i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0594 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0594 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0594 - 0.998i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.0594 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.115376518\)
\(L(\frac12)\) \(\approx\) \(1.115376518\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (1.01 - 3.15i)T \)
good5 \( 1 + (0.943 - 1.29i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (1.22 + 1.68i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.11 - 2.98i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.93 - 2.25i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.571iT - 23T^{2} \)
29 \( 1 + (0.282 - 0.867i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.53 - 4.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0690 - 0.212i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.429 - 1.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.30iT - 43T^{2} \)
47 \( 1 + (2.53 - 0.824i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.49 - 2.05i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.53 - 0.823i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.63 - 6.37i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.07T + 67T^{2} \)
71 \( 1 + (-1.60 + 2.20i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.42 + 2.73i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.22 + 1.68i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.76 - 2.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.76iT - 89T^{2} \)
97 \( 1 + (-5.45 + 3.96i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.18185636324290362070113897441, −9.120984557079678046140401611378, −8.008133763319966398965409407576, −7.29842614126222936441984005193, −6.45679638694081614072481868262, −5.58360687796028840185427310273, −4.56510117710153363046584686554, −3.69257326470597015287239233560, −2.85550543024365554700987814252, −1.67575784894923427955463536728, 0.35572040442494908016255087809, 2.29812367330958392121215707172, 3.42375694292728071402300619119, 4.30131947512170763561087435581, 5.18902018813257575016904029053, 5.97555816863644079787613133357, 6.85869792586186547265186461833, 7.71594456538182872769636044721, 8.517842544782502806780530817329, 9.116089149880477607319963360097

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