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Properties

Label 2-1386-33.8-c1-0-13
Degree $2$
Conductor $1386$
Sign $0.398 + 0.917i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−1.15 + 1.59i)5-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + 1.96i·10-s + (−0.128 − 3.31i)11-s + (1.23 + 1.69i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (2.55 + 1.85i)17-s + (4.09 − 1.33i)19-s + (1.15 + 1.59i)20-s + (−2.05 − 2.60i)22-s − 8.25i·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.517 + 0.711i)5-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + 0.622i·10-s + (−0.0386 − 0.999i)11-s + (0.341 + 0.470i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.619 + 0.449i)17-s + (0.940 − 0.305i)19-s + (0.258 + 0.355i)20-s + (−0.437 − 0.555i)22-s − 1.72i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.002797696\)
\(L(\frac12)\) \(\approx\) \(2.002797696\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.128 + 3.31i)T \)
good5 \( 1 + (1.15 - 1.59i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.23 - 1.69i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.55 - 1.85i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.09 + 1.33i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.25iT - 23T^{2} \)
29 \( 1 + (-1.44 + 4.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.02 + 5.10i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.68 - 5.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0464 + 0.142i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 + (-3.08 + 1.00i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.670 + 0.923i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (13.4 + 4.38i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.45 + 8.88i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 15.7T + 67T^{2} \)
71 \( 1 + (7.04 - 9.69i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.55 - 1.15i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.69 - 3.70i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.7 - 7.81i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.06iT - 89T^{2} \)
97 \( 1 + (4.98 - 3.62i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.650322747161272454459565146439, −8.567844125368706936565826147781, −7.81962116352947310971615835886, −6.69839458955593688410241325830, −6.24053578478985431269270555175, −5.16583527455912704221263621852, −4.05340899159679232857228522193, −3.34040517712946881595169625828, −2.50284256268603967474460884207, −0.78556608286340176719677641406, 1.25673990300817320995966868642, 2.92549369695368262749803678156, 3.76390835128931070191800684546, 4.82855259496069197031414916076, 5.37666804855349075950838040920, 6.39316551065807577848517144686, 7.44163422631575295594402021497, 7.82802931927147383345898431435, 8.873127441700782170115213322586, 9.606982177258562268308000814846

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