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Properties

Label 2-1386-33.17-c1-0-9
Degree $2$
Conductor $1386$
Sign $-0.744 - 0.667i$
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 + (3.30 + 1.07i)5-s + (−0.587 − 0.809i)7-s + (−0.809 − 0.587i)8-s + 3.47i·10-s + (−0.456 + 3.28i)11-s + (−4.44 + 1.44i)13-s + (0.587 − 0.809i)14-s + (0.309 − 0.951i)16-s + (0.244 − 0.751i)17-s + (−2.58 + 3.55i)19-s + (−3.30 + 1.07i)20-s + (−3.26 + 0.581i)22-s + 1.98i·23-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (1.47 + 0.480i)5-s + (−0.222 − 0.305i)7-s + (−0.286 − 0.207i)8-s + 1.09i·10-s + (−0.137 + 0.990i)11-s + (−1.23 + 0.400i)13-s + (0.157 − 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.0592 − 0.182i)17-s + (−0.592 + 0.815i)19-s + (−0.739 + 0.240i)20-s + (−0.696 + 0.123i)22-s + 0.414i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.744 - 0.667i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.817152459\)
\(L(\frac12)\) \(\approx\) \(1.817152459\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.456 - 3.28i)T \)
good5 \( 1 + (-3.30 - 1.07i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (4.44 - 1.44i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.244 + 0.751i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.58 - 3.55i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.98iT - 23T^{2} \)
29 \( 1 + (4.94 - 3.59i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.57 - 4.84i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.78 + 1.29i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.36 - 4.62i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.15iT - 43T^{2} \)
47 \( 1 + (-2.53 + 3.48i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.63 + 1.18i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.54 + 10.3i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-13.9 - 4.52i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + (2.32 + 0.755i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.79 - 7.98i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (11.5 - 3.76i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.07 - 3.29i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.827iT - 89T^{2} \)
97 \( 1 + (3.84 + 11.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.758292494633591533839515889905, −9.343804277747060197539485677557, −8.129743879314583699995363591416, −7.09737281218903254748165511853, −6.77654460223237888256417251290, −5.74206133893348657068813355643, −5.09494865561344545951170606647, −4.07497457076494184529178566827, −2.72215726741823462466350600740, −1.79376964958540577582244873370, 0.65024651951239474136132595940, 2.19969341051228250134265605347, 2.68288426399405419689847991674, 4.11658294143166107854985033910, 5.20236694062290509919163580023, 5.72303105660717881852119154611, 6.50199045971763284300032848596, 7.79668426087235784741991817696, 8.842938447723168905897670689207, 9.352501764415893760215620198102

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