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Properties

Label 2-1386-33.17-c1-0-4
Degree $2$
Conductor $1386$
Sign $0.230 - 0.972i$
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 + (−0.678 − 0.220i)5-s + (0.587 + 0.809i)7-s + (0.809 + 0.587i)8-s + 0.713i·10-s + (2.13 + 2.53i)11-s + (−2.85 + 0.929i)13-s + (0.587 − 0.809i)14-s + (0.309 − 0.951i)16-s + (1.78 − 5.49i)17-s + (−3.17 + 4.37i)19-s + (0.678 − 0.220i)20-s + (1.75 − 2.81i)22-s + 1.21i·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.303 − 0.0986i)5-s + (0.222 + 0.305i)7-s + (0.286 + 0.207i)8-s + 0.225i·10-s + (0.644 + 0.764i)11-s + (−0.793 + 0.257i)13-s + (0.157 − 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.432 − 1.33i)17-s + (−0.729 + 1.00i)19-s + (0.151 − 0.0493i)20-s + (0.373 − 0.600i)22-s + 0.254i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.230 - 0.972i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.230 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7589864740\)
\(L(\frac12)\) \(\approx\) \(0.7589864740\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.13 - 2.53i)T \)
good5 \( 1 + (0.678 + 0.220i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (2.85 - 0.929i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.78 + 5.49i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.17 - 4.37i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.21iT - 23T^{2} \)
29 \( 1 + (3.65 - 2.65i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.763 - 2.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.85 - 6.43i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.05 - 0.769i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 + (-5.21 + 7.17i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.54 - 2.45i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.51 - 3.46i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.8 - 3.86i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (-8.42 - 2.73i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.32 + 5.94i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-7.16 + 2.32i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.54 - 14.0i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 + (-5.26 - 16.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.743520815012778530116154746795, −9.142115733496467391453000394365, −8.210369602761061150735041442925, −7.46275195567638445892527974935, −6.62141322717382476867141790730, −5.34359015893877145329138966002, −4.56816970552638549488682924824, −3.66292248534895835619350914273, −2.49789831327330117081543951247, −1.46823179102177789871347475231, 0.33691098777627263121873011900, 1.95508840669961054861815990086, 3.54062241861057123393656008371, 4.27115395062970063948342926898, 5.40744344511174897187818019472, 6.14533879536424764252745485148, 7.07323615265568220388439157359, 7.72811019214004798236282900398, 8.558587748574422790750917591780, 9.179493982049605874581467116116

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