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Properties

Label 2-1386-33.17-c1-0-8
Degree $2$
Conductor $1386$
Sign $0.300 - 0.953i$
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 + (−3.13 − 1.01i)5-s + (−0.587 − 0.809i)7-s + (−0.809 − 0.587i)8-s − 3.29i·10-s + (−3.17 + 0.943i)11-s + (6.67 − 2.17i)13-s + (0.587 − 0.809i)14-s + (0.309 − 0.951i)16-s + (0.210 − 0.649i)17-s + (−3.84 + 5.29i)19-s + (3.13 − 1.01i)20-s + (−1.88 − 2.73i)22-s + 1.86i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (−1.40 − 0.455i)5-s + (−0.222 − 0.305i)7-s + (−0.286 − 0.207i)8-s − 1.04i·10-s + (−0.958 + 0.284i)11-s + (1.85 − 0.601i)13-s + (0.157 − 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.0511 − 0.157i)17-s + (−0.882 + 1.21i)19-s + (0.700 − 0.227i)20-s + (−0.400 − 0.582i)22-s + 0.388i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.300 - 0.953i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.133675942\)
\(L(\frac12)\) \(\approx\) \(1.133675942\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.17 - 0.943i)T \)
good5 \( 1 + (3.13 + 1.01i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-6.67 + 2.17i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.210 + 0.649i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.84 - 5.29i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.86iT - 23T^{2} \)
29 \( 1 + (-5.86 + 4.25i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.831 - 2.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.03 + 5.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.41 - 6.11i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.19iT - 43T^{2} \)
47 \( 1 + (1.15 - 1.58i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (10.4 - 3.38i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.48 - 8.93i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.89 - 3.21i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.29T + 67T^{2} \)
71 \( 1 + (0.0882 + 0.0286i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.91 - 9.51i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-9.12 + 2.96i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.68 + 11.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.71iT - 89T^{2} \)
97 \( 1 + (3.73 + 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.626857655060218686973148317842, −8.453001864894907174861088214100, −8.084108248958683825063065360451, −7.57514970881004851785001580403, −6.37405070143222357902839005026, −5.72119984481136289437132034929, −4.47912829227362562364420376846, −3.98615620732544776788445105649, −3.00341870166115053749476671151, −0.910496138127907529102450766404, 0.61044579673114467211436642403, 2.38609433613000785798399875310, 3.34571412529128303042267910521, 4.05052313564451306694718436432, 4.94565787002014589108990848198, 6.19485952047389565239043827760, 6.85985079936022883967270032090, 8.130900131907479412690097565705, 8.467498726823710924854768267809, 9.388698800198080248577108552421

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