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Properties

Label 2-1386-33.17-c1-0-20
Degree $2$
Conductor $1386$
Sign $-0.778 + 0.627i$
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 + (−0.411 − 0.133i)5-s + (−0.587 − 0.809i)7-s + (0.809 + 0.587i)8-s + 0.432i·10-s + (0.137 + 3.31i)11-s + (−1.34 + 0.436i)13-s + (−0.587 + 0.809i)14-s + (0.309 − 0.951i)16-s + (0.223 − 0.688i)17-s + (3.54 − 4.87i)19-s + (0.411 − 0.133i)20-s + (3.10 − 1.15i)22-s − 1.67i·23-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.184 − 0.0598i)5-s + (−0.222 − 0.305i)7-s + (0.286 + 0.207i)8-s + 0.136i·10-s + (0.0414 + 0.999i)11-s + (−0.372 + 0.121i)13-s + (−0.157 + 0.216i)14-s + (0.0772 − 0.237i)16-s + (0.0542 − 0.166i)17-s + (0.812 − 1.11i)19-s + (0.0920 − 0.0299i)20-s + (0.662 − 0.246i)22-s − 0.348i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.778 + 0.627i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8678834838\)
\(L(\frac12)\) \(\approx\) \(0.8678834838\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.137 - 3.31i)T \)
good5 \( 1 + (0.411 + 0.133i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.34 - 0.436i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.223 + 0.688i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.54 + 4.87i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.67iT - 23T^{2} \)
29 \( 1 + (0.367 - 0.266i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.99 + 6.15i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.85 + 4.98i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.35 + 0.985i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.52iT - 43T^{2} \)
47 \( 1 + (2.88 - 3.97i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.253 - 0.0824i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.56 + 4.90i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.756 - 0.245i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 0.680T + 67T^{2} \)
71 \( 1 + (2.38 + 0.774i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.78 + 5.20i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.90 - 1.92i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.83 - 5.64i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 + (5.31 + 16.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.642701304559829866394129295006, −8.603589784486534141720180978162, −7.56129766941121379213448360909, −7.11943206035059228804400083348, −5.88922714490599444672720633876, −4.73757428885240699301908835170, −4.11971070731648641957511357769, −2.92973714045103573197875220066, −1.95445257754415349398617112876, −0.40371484642263954426690998955, 1.33936749953110952684946813178, 3.00608725563344206500052700383, 3.87365943476986200683836294713, 5.15102814394013447188341093525, 5.81313722554268249944659834743, 6.57873542480137205307478108797, 7.63120067572277628294748570454, 8.120447178592140385667244687007, 9.030390755942253235443429089708, 9.737545569626063378892213818250

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