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Properties

Label 2-1386-33.17-c1-0-16
Degree $2$
Conductor $1386$
Sign $0.697 + 0.716i$
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 + (2.17 + 0.708i)5-s + (0.587 + 0.809i)7-s + (0.809 + 0.587i)8-s − 2.29i·10-s + (0.675 − 3.24i)11-s + (3.85 − 1.25i)13-s + (0.587 − 0.809i)14-s + (0.309 − 0.951i)16-s + (0.654 − 2.01i)17-s + (−2.04 + 2.81i)19-s + (−2.17 + 0.708i)20-s + (−3.29 + 0.360i)22-s + 4.13i·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.974 + 0.316i)5-s + (0.222 + 0.305i)7-s + (0.286 + 0.207i)8-s − 0.724i·10-s + (0.203 − 0.979i)11-s + (1.06 − 0.347i)13-s + (0.157 − 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.158 − 0.488i)17-s + (−0.469 + 0.646i)19-s + (−0.487 + 0.158i)20-s + (−0.702 + 0.0768i)22-s + 0.863i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.697 + 0.716i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.873686541\)
\(L(\frac12)\) \(\approx\) \(1.873686541\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.675 + 3.24i)T \)
good5 \( 1 + (-2.17 - 0.708i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (-3.85 + 1.25i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.654 + 2.01i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.04 - 2.81i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.13iT - 23T^{2} \)
29 \( 1 + (-5.33 + 3.87i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.408 + 1.25i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.69 - 4.86i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.63 - 6.27i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 + (-4.08 + 5.62i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.58 + 2.79i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.63 - 2.24i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.23 - 1.37i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 + (-9.93 - 3.22i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.47 - 7.53i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (14.1 - 4.60i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.27 - 7.01i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (3.69 + 11.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.694466061686269741228688490315, −8.597902672406026516714598292437, −8.306031014236385572648897046411, −6.98890375351282901945137260230, −5.92574270285914923099150623746, −5.54018057793911072377843642196, −4.11322091093136531669301840858, −3.17871120629812877510383108689, −2.19913154843470588189966614261, −1.05233962952526029186907204950, 1.17908884372387047969066666166, 2.24467360011539754347744176680, 3.92342124862859512529622557541, 4.73891716738165777931262764847, 5.65631489654997809010306769449, 6.46411834380878594754000114904, 7.09015670514003518875192281419, 8.138572323999696829553182015336, 8.955234473800413850641562819179, 9.399348725799159093884350234739

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