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Properties

Label 2-1386-21.17-c1-0-2
Degree $2$
Conductor $1386$
Sign $-0.179 - 0.983i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.179 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.127066490\)
\(L(\frac12)\) \(\approx\) \(1.127066490\)
\(L(\frac{3}{2})\) 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

−10.01772047117352641355726065894, −8.980256807722077101895185172203, −8.405109250709314342119736405935, −7.09154639365461998302493089981, −6.23069672694980414834655833103, −5.91114928695875603145499600224, −4.46880486145733902555076361930, −3.89003904204857789160609189564, −2.68764283275505860432709896208, −1.81729294404640062575683793678, 0.34096500590203714435901025834, 2.38843680633278269218322250730, 3.25931184494520688420644660879, 4.31258194977400333146966509720, 5.13076609332195538036530827007, 6.06062983950805910701627059045, 6.88846969009048991658172578897, 7.45799724479496484162584435394, 8.548669911905882914677205423023, 9.237631023432574345580598529520

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