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Properties

Label 2-1386-33.8-c1-0-15
Degree $2$
Conductor $1386$
Sign $0.478 + 0.878i$
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 + (1.24 − 1.70i)5-s + (0.951 + 0.309i)7-s + (0.309 + 0.951i)8-s + 2.11i·10-s + (−0.422 − 3.28i)11-s + (−0.352 − 0.485i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (0.694 + 0.504i)17-s + (2.22 − 0.721i)19-s + (−1.24 − 1.70i)20-s + (2.27 + 2.41i)22-s − 2.92i·23-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (0.555 − 0.764i)5-s + (0.359 + 0.116i)7-s + (0.109 + 0.336i)8-s + 0.668i·10-s + (−0.127 − 0.991i)11-s + (−0.0978 − 0.134i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.168 + 0.122i)17-s + (0.509 − 0.165i)19-s + (−0.277 − 0.382i)20-s + (0.485 + 0.514i)22-s − 0.610i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.478 + 0.878i$
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: \((2,\ 1386,\ (\ :1/2),\ 0.478 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.309556820\)
\(L(\frac12)\) \(\approx\) \(1.309556820\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.422 + 3.28i)T \)
good5 \( 1 + (-1.24 + 1.70i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.352 + 0.485i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.694 - 0.504i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.22 + 0.721i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.92iT - 23T^{2} \)
29 \( 1 + (-1.88 + 5.81i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.894 + 0.649i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.37 - 7.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.44 + 10.6i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.69iT - 43T^{2} \)
47 \( 1 + (5.90 - 1.91i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.49 + 2.05i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.201 - 0.0653i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.31 + 10.0i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 + (-2.88 + 3.96i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.80 - 0.585i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.03 + 6.92i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.58 + 4.78i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-5.61 + 4.07i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.378205687422738796829621178575, −8.459165040061189211404041910813, −8.173729605528136129977183556651, −7.03060240589334226361340563608, −6.08770373090590610842994605798, −5.40306810396394324345362141345, −4.63186358001829918455464111802, −3.19115318756308004098943657052, −1.86816090825829420211914789431, −0.67787328852214306553424419515, 1.43865850183962914799611196843, 2.42194329566808413465494040792, 3.39260168498466901640915456080, 4.59250427700207278423563373972, 5.57152214387950026855671589519, 6.74616475549317518032394370681, 7.25466524420887197095454327229, 8.128692701055414319911615402598, 9.073446207310954075983793221428, 9.919153710730956825420928111413

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