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Properties

Label 2-1386-33.2-c1-0-17
Degree $2$
Conductor $1386$
Sign $-0.890 + 0.455i$
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 + (−1.69 + 0.550i)5-s + (−0.587 + 0.809i)7-s + (−0.809 + 0.587i)8-s + 1.78i·10-s + (2.97 + 1.47i)11-s + (0.352 + 0.114i)13-s + (0.587 + 0.809i)14-s + (0.309 + 0.951i)16-s + (−1.79 − 5.51i)17-s + (−0.665 − 0.915i)19-s + (1.69 + 0.550i)20-s + (2.32 − 2.36i)22-s − 5.35i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.758 + 0.246i)5-s + (−0.222 + 0.305i)7-s + (−0.286 + 0.207i)8-s + 0.563i·10-s + (0.895 + 0.444i)11-s + (0.0977 + 0.0317i)13-s + (0.157 + 0.216i)14-s + (0.0772 + 0.237i)16-s + (−0.434 − 1.33i)17-s + (−0.152 − 0.210i)19-s + (0.379 + 0.123i)20-s + (0.494 − 0.505i)22-s − 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8652719674\)
\(L(\frac12)\) \(\approx\) \(0.8652719674\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.97 - 1.47i)T \)
good5 \( 1 + (1.69 - 0.550i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.352 - 0.114i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.79 + 5.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.665 + 0.915i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.35iT - 23T^{2} \)
29 \( 1 + (6.33 + 4.60i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.14 - 5.19i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.50 + 3.27i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.57iT - 43T^{2} \)
47 \( 1 + (7.43 + 10.2i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (11.8 + 3.83i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.16 - 11.2i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.20 - 2.33i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.02T + 67T^{2} \)
71 \( 1 + (-7.04 + 2.28i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.87 - 9.45i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.62 - 1.82i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.46 + 7.58i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.106iT - 89T^{2} \)
97 \( 1 + (-5.52 + 17.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.375783412894319213016287161932, −8.612174794120638428661659674918, −7.58775262201321072500791019381, −6.79871970693152284020812411972, −5.88656916490621423132869638705, −4.68532704404957774351713294242, −4.05191620698404270811201327064, −3.04489504024869040767707304995, −2.02484198191796962383374958522, −0.34034207783349861681675987748, 1.43045814509246157930691567167, 3.34453846703823581361623427674, 3.96294768424809729606099570559, 4.79055447060025135112893766756, 6.09067350930202139316922088999, 6.42888665223150144361177223065, 7.78280215376400318287414301743, 7.923891959693107087041697049046, 9.113041011494945861542410204497, 9.555180063634029970575853552283

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