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Properties

Label 2-1386-33.17-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.616 - 0.787i$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (1.68 + 0.547i)5-s + (−0.587 − 0.809i)7-s + (−0.809 − 0.587i)8-s + 1.77i·10-s + (0.882 + 3.19i)11-s + (5.61 − 1.82i)13-s + (0.587 − 0.809i)14-s + (0.309 − 0.951i)16-s + (2.35 − 7.23i)17-s + (2.36 − 3.25i)19-s + (−1.68 + 0.547i)20-s + (−2.76 + 1.82i)22-s + 5.31i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.752 + 0.244i)5-s + (−0.222 − 0.305i)7-s + (−0.286 − 0.207i)8-s + 0.559i·10-s + (0.266 + 0.963i)11-s + (1.55 − 0.506i)13-s + (0.157 − 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.570 − 1.75i)17-s + (0.543 − 0.747i)19-s + (−0.376 + 0.122i)20-s + (−0.590 + 0.389i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.616 - 0.787i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.217023998\)
\(L(\frac12)\) \(\approx\) \(2.217023998\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.882 - 3.19i)T \)
good5 \( 1 + (-1.68 - 0.547i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-5.61 + 1.82i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.35 + 7.23i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.36 + 3.25i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.31iT - 23T^{2} \)
29 \( 1 + (-1.43 + 1.04i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.558 + 1.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.24 - 4.53i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.68 - 4.13i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.27iT - 43T^{2} \)
47 \( 1 + (1.37 - 1.89i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.89 + 3.21i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.96 - 8.21i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.06 + 1.32i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.57T + 67T^{2} \)
71 \( 1 + (-8.10 - 2.63i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.05 + 8.33i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.89 - 1.91i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.63 + 5.02i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.93iT - 89T^{2} \)
97 \( 1 + (1.89 + 5.82i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.634019260822928431140107987170, −8.982310551218946804933140864224, −7.85352410688645035203638361994, −7.18470702885488054692228344171, −6.42167970869056790092231363961, −5.61793999004181884348913531855, −4.83007520692377188214681128529, −3.69602301260016075841564659624, −2.72649596225601582996349748141, −1.13796687729625802804229892869, 1.15603170064482167953997752904, 2.06869624212334235737261257207, 3.51602258330700060584856975521, 3.95239493056225343067098370117, 5.59908447293629485766445194052, 5.80732190739917001485740261346, 6.73010615248091511655835829790, 8.344219593697397208562468357585, 8.642036581209971534281426874908, 9.490133447331213928401907946672

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