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Properties

Label 2-1386-21.5-c1-0-21
Degree $2$
Conductor $1386$
Sign $-0.867 - 0.497i$
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 + (−0.171 + 0.296i)5-s + (−2.33 − 1.23i)7-s − 0.999i·8-s + (0.296 − 0.171i)10-s + (0.866 − 0.5i)11-s − 4.10i·13-s + (1.40 + 2.23i)14-s + (−0.5 + 0.866i)16-s + (2.61 + 4.52i)17-s + (−6.70 − 3.86i)19-s − 0.342·20-s − 0.999·22-s + (1.74 + 1.00i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.0766 + 0.132i)5-s + (−0.884 − 0.466i)7-s − 0.353i·8-s + (0.0939 − 0.0542i)10-s + (0.261 − 0.150i)11-s − 1.13i·13-s + (0.376 + 0.598i)14-s + (−0.125 + 0.216i)16-s + (0.633 + 1.09i)17-s + (−1.53 − 0.887i)19-s − 0.0766·20-s − 0.213·22-s + (0.363 + 0.209i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.867 - 0.497i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02136474735\)
\(L(\frac12)\) \(\approx\) \(0.02136474735\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 + 1.23i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.171 - 0.296i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
17 \( 1 + (-2.61 - 4.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.70 + 3.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.74 - 1.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.958iT - 29T^{2} \)
31 \( 1 + (1.01 - 0.585i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.314 + 0.545i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.34T + 41T^{2} \)
43 \( 1 + 6.93T + 43T^{2} \)
47 \( 1 + (5.28 - 9.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.37 - 3.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.67 + 9.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.21 + 4.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.84 + 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.47iT - 71T^{2} \)
73 \( 1 + (7.35 - 4.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.08T + 83T^{2} \)
89 \( 1 + (-0.164 + 0.284i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.22iT - 97T^{2} \)
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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

−9.185307512722021957341618377173, −8.365488848831215519704864575770, −7.62069581098770039642126660763, −6.68513257272144363828814558990, −6.05861682874374057969456198551, −4.76058712089812563424549509263, −3.54478268873331023697300109298, −2.98560545392302649837947293806, −1.46321687632258450971320067758, −0.01061988388747111380636292519, 1.72994548986252362621135729564, 2.89548500794021515853690388012, 4.13452459033276222237992519271, 5.13979944158350437086952345051, 6.30770995030425784301296996089, 6.64671160396935849955358172015, 7.63564014418658609675130509791, 8.673394108758871848583675724975, 9.064980490282893544590152895224, 9.972055032347093596203578246529

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