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Properties

Label 2-1386-21.5-c1-0-15
Degree $2$
Conductor $1386$
Sign $0.503 + 0.864i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.474 + 0.821i)5-s + (1.41 + 2.23i)7-s − 0.999i·8-s + (0.821 − 0.474i)10-s + (0.866 − 0.5i)11-s − 7.03i·13-s + (−0.102 − 2.64i)14-s + (−0.5 + 0.866i)16-s + (−2.29 − 3.98i)17-s + (1.93 + 1.11i)19-s − 0.948·20-s − 0.999·22-s + (−7.17 − 4.14i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.212 + 0.367i)5-s + (0.533 + 0.846i)7-s − 0.353i·8-s + (0.259 − 0.150i)10-s + (0.261 − 0.150i)11-s − 1.94i·13-s + (−0.0272 − 0.706i)14-s + (−0.125 + 0.216i)16-s + (−0.557 − 0.966i)17-s + (0.443 + 0.256i)19-s − 0.212·20-s − 0.213·22-s + (−1.49 − 0.864i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.503 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.126131307\)
\(L(\frac12)\) \(\approx\) \(1.126131307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.41 - 2.23i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.474 - 0.821i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 7.03iT - 13T^{2} \)
17 \( 1 + (2.29 + 3.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.93 - 1.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.17 + 4.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.930iT - 29T^{2} \)
31 \( 1 + (2.80 - 1.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.57 + 7.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.54T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + (-0.251 + 0.435i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.22 + 4.75i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.02 + 1.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.94 + 5.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-4.85 + 2.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.16 - 7.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (0.441 - 0.764i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−9.359683859591619390164689349531, −8.723760336897935010738211287117, −7.81370531976763173337953887593, −7.40076415982817217370629086433, −6.04674204555893194631526700434, −5.43085907759358478290013488636, −4.16380512232066305616656414339, −2.99544011483110752630343837979, −2.27697426822747092533939026634, −0.64767204884392853600923407409, 1.17474032720256580952115995433, 2.17040123523075815459528257845, 4.16394077991032753674357143654, 4.30948081478887690850977419584, 5.75360295360247528756710166508, 6.62358714811623685187109372012, 7.35100347148700730317766077944, 8.095090813994137332165953950171, 8.898799706514655415543437422008, 9.560498900290030814838630050145

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