Properties

Degree $2$
Conductor $1386$
Sign $0.729 + 0.683i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s − 4.23·5-s + (−2.59 + 0.531i)7-s + i·8-s + 4.23i·10-s + i·11-s + (0.531 + 2.59i)14-s + 16-s − 5.29·17-s − 0.250i·19-s + 4.23·20-s + 22-s − 4.50i·23-s + 12.9·25-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.89·5-s + (−0.979 + 0.200i)7-s + 0.353i·8-s + 1.33i·10-s + 0.301i·11-s + (0.142 + 0.692i)14-s + 0.250·16-s − 1.28·17-s − 0.0573i·19-s + 0.946·20-s + 0.213·22-s − 0.938i·23-s + 2.58·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.729 + 0.683i$
Motivic weight: \(1\)
Character: $\chi_{1386} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6039525101\)
\(L(\frac12)\) \(\approx\) \(0.6039525101\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (2.59 - 0.531i)T \)
11 \( 1 - iT \)
good5 \( 1 + 4.23T + 5T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 0.250iT - 19T^{2} \)
23 \( 1 + 4.50iT - 23T^{2} \)
29 \( 1 + 1.05iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 - 2.50T + 37T^{2} \)
41 \( 1 + 1.31T + 41T^{2} \)
43 \( 1 - 7.18T + 43T^{2} \)
47 \( 1 - 7.15T + 47T^{2} \)
53 \( 1 - 4.68iT - 53T^{2} \)
59 \( 1 + 8.46T + 59T^{2} \)
61 \( 1 + 14.8iT - 61T^{2} \)
67 \( 1 - 9.42T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 7.15iT - 73T^{2} \)
79 \( 1 + 0.377T + 79T^{2} \)
83 \( 1 - 1.84T + 83T^{2} \)
89 \( 1 + 1.56T + 89T^{2} \)
97 \( 1 - 7.96iT - 97T^{2} \)
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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

−9.366062562148922165077246433183, −8.780386412323091173716483351104, −7.987157843236984875627122921795, −7.10219404902562560306357823363, −6.36530817148919665587401063543, −4.86238815092054103728211238531, −4.18090228637267515331927433372, −3.40158138810326586656633225595, −2.48558810479706413478944954104, −0.54415978433004949756262414996, 0.53318983015303867568217978450, 2.89500602328392138097051609505, 3.91985716700598708460266811801, 4.33052474427267058349115871436, 5.62773122559778845804791871125, 6.62861004401738990838720381083, 7.28399987463288950569886797736, 7.88760394610849326714201223031, 8.755157544583460400167476419359, 9.367342622223864787889463051843

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