↑

Properties

Label 2-2e4-4.3-c10-0-0
Degree $2$
Conductor $16$
Sign $-0.500 - 0.866i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 214. i·3-s − 3.26e3·5-s + 5.80e3i·7-s + 1.30e4·9-s + 2.71e5i·11-s − 5.57e5·13-s + 6.99e5i·15-s − 7.95e5·17-s + 4.01e6i·19-s + 1.24e6·21-s − 7.93e6i·23-s + 8.92e5·25-s − 1.54e7i·27-s + 7.81e6·29-s − 1.15e7i·31-s + ⋯
L(s)  = 1  − 0.882i·3-s − 1.04·5-s + 0.345i·7-s + 0.221·9-s + 1.68i·11-s − 1.50·13-s + 0.921i·15-s − 0.560·17-s + 1.62i·19-s + 0.304·21-s − 1.23i·23-s + 0.0913·25-s − 1.07i·27-s + 0.380·29-s − 0.403i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.500 - 0.866i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.500 - 0.866i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.204707 + 0.354563i\)
\(L(\frac12)\) \(\approx\) \(0.204707 + 0.354563i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 214. iT - 5.90e4T^{2} \)
5 \( 1 + 3.26e3T + 9.76e6T^{2} \)
7 \( 1 - 5.80e3iT - 2.82e8T^{2} \)
11 \( 1 - 2.71e5iT - 2.59e10T^{2} \)
13 \( 1 + 5.57e5T + 1.37e11T^{2} \)
17 \( 1 + 7.95e5T + 2.01e12T^{2} \)
19 \( 1 - 4.01e6iT - 6.13e12T^{2} \)
23 \( 1 + 7.93e6iT - 4.14e13T^{2} \)
29 \( 1 - 7.81e6T + 4.20e14T^{2} \)
31 \( 1 + 1.15e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.07e8T + 4.80e15T^{2} \)
41 \( 1 + 5.82e7T + 1.34e16T^{2} \)
43 \( 1 - 7.58e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.19e8iT - 5.25e16T^{2} \)
53 \( 1 + 4.68e8T + 1.74e17T^{2} \)
59 \( 1 + 8.13e8iT - 5.11e17T^{2} \)
61 \( 1 + 4.39e8T + 7.13e17T^{2} \)
67 \( 1 - 1.14e9iT - 1.82e18T^{2} \)
71 \( 1 - 9.19e6iT - 3.25e18T^{2} \)
73 \( 1 + 8.89e8T + 4.29e18T^{2} \)
79 \( 1 - 5.96e9iT - 9.46e18T^{2} \)
83 \( 1 + 2.00e8iT - 1.55e19T^{2} \)
89 \( 1 - 7.64e9T + 3.11e19T^{2} \)
97 \( 1 + 6.43e9T + 7.37e19T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.30056017659169056970151707367, −15.63264426413728961230109419769, −14.55352604477001767903157294022, −12.48532577753635673357753047248, −12.16660310188118915971574994012, −9.998594944504876633032118969232, −7.927923402745418275615064704655, −6.93794030710311867454845399939, −4.48437215658643818055072984822, −2.03607356497616562148237030664, 0.18548361289326149450554229332, 3.40870268209642714033529020828, 4.85806993576184713190034775078, 7.28773091429851534283290942161, 8.986285561238946826011777779989, 10.60953430757752636473749483221, 11.76308808771619221251441585596, 13.61870875181866701978872439765, 15.26366170390198409240410871506, 15.99064584616877290061427672631

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