Properties

Label 2-1386-33.8-c1-0-3
Degree $2$
Conductor $1386$
Sign $-0.567 - 0.823i$
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 + (0.0348 − 0.0479i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s + 0.0592i·10-s + (0.764 + 3.22i)11-s + (−2.76 − 3.80i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (0.0206 + 0.0149i)17-s + (−5.33 + 1.73i)19-s + (−0.0348 − 0.0479i)20-s + (−2.51 − 2.16i)22-s + 7.57i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.0155 − 0.0214i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s + 0.0187i·10-s + (0.230 + 0.973i)11-s + (−0.765 − 1.05i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.00499 + 0.00363i)17-s + (−1.22 + 0.397i)19-s + (−0.00779 − 0.0107i)20-s + (−0.536 − 0.460i)22-s + 1.57i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.567 - 0.823i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8466879833\)
\(L(\frac12)\) \(\approx\) \(0.8466879833\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.764 - 3.22i)T \)
good5 \( 1 + (-0.0348 + 0.0479i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.76 + 3.80i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0206 - 0.0149i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.33 - 1.73i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.57iT - 23T^{2} \)
29 \( 1 + (1.03 - 3.17i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.61 + 2.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.34 + 4.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.884 + 2.72i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 + (6.78 - 2.20i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.47 - 7.53i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.52 - 0.821i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.10 + 1.52i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.35T + 67T^{2} \)
71 \( 1 + (8.20 - 11.2i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.975 - 0.316i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.60 - 10.4i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-13.6 - 9.95i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.58iT - 89T^{2} \)
97 \( 1 + (13.5 - 9.81i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.728065488858708752987836222315, −9.113100071417796929985269661070, −8.105490461625468684692559693100, −7.53394804878213790305415663875, −6.80011983668374261913407255993, −5.69035759162504198405670920926, −5.04678398885629011988441524025, −3.93874671815685696095933815687, −2.54117430344791805679367795919, −1.41769927352209248508749940597, 0.42081855609270460125368702001, 1.94931969008815241108346855672, 2.86296249619845513245308548477, 4.17722809908582148919335929349, 4.82201036437942206787351027493, 6.36156805651041302057452586319, 6.72824558168503128627732598183, 7.987261447085762327186481450973, 8.556551853505489216991266104029, 9.194795997735966787883818700695

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