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Properties

Label 2-1386-21.5-c1-0-9
Degree $2$
Conductor $1386$
Sign $-0.219 - 0.975i$
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 + (−0.645 + 1.11i)5-s + (2.63 + 0.221i)7-s + 0.999i·8-s + (−1.11 + 0.645i)10-s + (−0.866 + 0.5i)11-s + 3.78i·13-s + (2.17 + 1.51i)14-s + (−0.5 + 0.866i)16-s + (−0.552 − 0.957i)17-s + (2.02 + 1.16i)19-s − 1.29·20-s − 0.999·22-s + (−2.83 − 1.63i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.288 + 0.500i)5-s + (0.996 + 0.0837i)7-s + 0.353i·8-s + (−0.353 + 0.204i)10-s + (−0.261 + 0.150i)11-s + 1.05i·13-s + (0.580 + 0.403i)14-s + (−0.125 + 0.216i)16-s + (−0.134 − 0.232i)17-s + (0.464 + 0.268i)19-s − 0.288·20-s − 0.213·22-s + (−0.591 − 0.341i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.219 - 0.975i$
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: \((2,\ 1386,\ (\ :1/2),\ -0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.339881438\)
\(L(\frac12)\) \(\approx\) \(2.339881438\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.221i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.645 - 1.11i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.78iT - 13T^{2} \)
17 \( 1 + (0.552 + 0.957i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 1.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.83 + 1.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.78iT - 29T^{2} \)
31 \( 1 + (-3.81 + 2.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.09 - 5.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.16T + 41T^{2} \)
43 \( 1 + 8.44T + 43T^{2} \)
47 \( 1 + (-1.74 + 3.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.02 - 1.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.30 - 3.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.60 - 0.924i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.79 - 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.96iT - 71T^{2} \)
73 \( 1 + (-4.98 + 2.87i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.49 + 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.95T + 83T^{2} \)
89 \( 1 + (1.79 - 3.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−9.858848487352556906934541612522, −8.764108434147595663279002928913, −8.084321835205283003267078807326, −7.22945820836719637344633618966, −6.64881402371968475064836378079, −5.54334501018982738454218874954, −4.75492560577568907303447308085, −3.96388063198847754289690824087, −2.83709415990672048917742229018, −1.69148521262291347174358438391, 0.800871826780587700238082587932, 2.09873638605500288000610281413, 3.27090096591063171229267912871, 4.29901397778346446029336362990, 5.06728310724921732787896993725, 5.71167598567515105550884292532, 6.84683828131523661330576395619, 7.974505119544422828337689164481, 8.279520260919286389940288043848, 9.436486316979475012758331879419

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