Properties

Label 2-1386-21.17-c1-0-5
Degree $2$
Conductor $1386$
Sign $0.345 - 0.938i$
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 + (1.22 + 2.12i)5-s + (−1 + 2.44i)7-s − 0.999i·8-s + (2.12 + 1.22i)10-s + (0.866 + 0.5i)11-s + 2.44i·13-s + (0.358 + 2.62i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 1.5i)17-s + (−1.5 + 0.866i)19-s + 2.44·20-s + 0.999·22-s + (−6.27 + 3.62i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.547 + 0.948i)5-s + (−0.377 + 0.925i)7-s − 0.353i·8-s + (0.670 + 0.387i)10-s + (0.261 + 0.150i)11-s + 0.679i·13-s + (0.0958 + 0.700i)14-s + (−0.125 − 0.216i)16-s + (−0.210 + 0.363i)17-s + (−0.344 + 0.198i)19-s + 0.547·20-s + 0.213·22-s + (−1.30 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.225924978\)
\(L(\frac12)\) \(\approx\) \(2.225924978\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-1.22 - 2.12i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (0.866 - 1.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.27 - 3.62i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.24iT - 29T^{2} \)
31 \( 1 + (-1.24 - 0.717i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-3.82 - 6.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.49 - 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.24 + 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.24iT - 71T^{2} \)
73 \( 1 + (-4.24 - 2.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (0.507 + 0.878i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.16iT - 97T^{2} \)
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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

−9.807414526869911262767971722931, −9.196560241530722531934249751254, −8.127929960132487953771493418139, −6.98965911318681141095237743320, −6.20250070466589216178332071695, −5.83168568156359700316301216585, −4.56500815012837048603909583352, −3.58905243063762032855672901465, −2.57299995655968140381479285024, −1.84771921106705036753927978461, 0.71355133495626385748004198988, 2.22470324613043723149188419874, 3.55567173492841204591039381589, 4.38337113959110430769144647826, 5.20900981303358921974494073848, 6.06787035259964825038864026023, 6.81732344845564740673954206088, 7.76892976311308975594212496784, 8.525119655310847488038445175514, 9.392082035088496312878530139872

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