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Properties

Label 2-1386-33.2-c1-0-15
Degree $2$
Conductor $1386$
Sign $0.435 + 0.900i$
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 + (2.76 − 0.897i)5-s + (0.587 − 0.809i)7-s + (−0.809 + 0.587i)8-s − 2.90i·10-s + (3.28 + 0.471i)11-s + (3.88 + 1.26i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (1.06 + 3.27i)17-s + (−0.0378 − 0.0520i)19-s + (−2.76 − 0.897i)20-s + (1.46 − 2.97i)22-s + 1.52i·23-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (1.23 − 0.401i)5-s + (0.222 − 0.305i)7-s + (−0.286 + 0.207i)8-s − 0.918i·10-s + (0.989 + 0.142i)11-s + (1.07 + 0.349i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.258 + 0.795i)17-s + (−0.00867 − 0.0119i)19-s + (−0.617 − 0.200i)20-s + (0.311 − 0.634i)22-s + 0.318i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.556858789\)
\(L(\frac12)\) \(\approx\) \(2.556858789\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 - 0.471i)T \)
good5 \( 1 + (-2.76 + 0.897i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-3.88 - 1.26i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.06 - 3.27i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0378 + 0.0520i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.52iT - 23T^{2} \)
29 \( 1 + (0.0763 + 0.0554i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.598 + 1.84i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.85 - 4.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.78 - 4.92i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.0378iT - 43T^{2} \)
47 \( 1 + (4.84 + 6.66i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.4 + 3.40i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.86 + 9.45i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.515 + 0.167i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.96T + 67T^{2} \)
71 \( 1 + (11.6 - 3.79i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.957 + 1.31i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.83 - 2.22i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.45 + 13.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.76iT - 89T^{2} \)
97 \( 1 + (1.89 - 5.82i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.674623212450004741467063296640, −8.799146319236739244025904281740, −8.114888654396031041199121155837, −6.63574409348009072708965719236, −6.11171956705810404842768957458, −5.17981003651913579367768602418, −4.21880389991794544160465294802, −3.34613834781664400396692576888, −1.85682024674308361150330835815, −1.29570385451585048764845459985, 1.32628872971959903917702207956, 2.66005715201911340369827592472, 3.72836059010402342373428425632, 4.88694246796900845110094650908, 5.83094477927010099048140955761, 6.26397071262370379510784639196, 7.07254818190438012918348175136, 8.120645074141298826024607818455, 8.970372443925443254313738681049, 9.502444390076238654370728866983

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