↑

Properties

Label 2-2e4-16.3-c10-0-0
Degree $2$
Conductor $16$
Sign $0.807 + 0.590i$
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  + (−19.3 + 25.4i)2-s + (−222. + 222. i)3-s + (−276. − 985. i)4-s + (−2.71e3 + 2.71e3i)5-s + (−1.36e3 − 9.95e3i)6-s − 2.48e4·7-s + (3.04e4 + 1.20e4i)8-s − 3.95e4i·9-s + (−1.67e4 − 1.21e5i)10-s + (1.58e5 + 1.58e5i)11-s + (2.80e5 + 1.57e5i)12-s + (−2.63e4 − 2.63e4i)13-s + (4.81e5 − 6.34e5i)14-s − 1.20e6i·15-s + (−8.95e5 + 5.45e5i)16-s + 1.46e6·17-s + ⋯
L(s)  = 1  + (−0.604 + 0.796i)2-s + (−0.913 + 0.913i)3-s + (−0.269 − 0.962i)4-s + (−0.867 + 0.867i)5-s + (−0.176 − 1.28i)6-s − 1.48·7-s + (0.930 + 0.366i)8-s − 0.669i·9-s + (−0.167 − 1.21i)10-s + (0.981 + 0.981i)11-s + (1.12 + 0.633i)12-s + (−0.0709 − 0.0709i)13-s + (0.894 − 1.18i)14-s − 1.58i·15-s + (−0.854 + 0.519i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.807 + 0.590i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.807 + 0.590i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0302331 - 0.00987478i\)
\(L(\frac12)\) \(\approx\) \(0.0302331 - 0.00987478i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (19.3 - 25.4i)T \)
good3 \( 1 + (222. - 222. i)T - 5.90e4iT^{2} \)
5 \( 1 + (2.71e3 - 2.71e3i)T - 9.76e6iT^{2} \)
7 \( 1 + 2.48e4T + 2.82e8T^{2} \)
11 \( 1 + (-1.58e5 - 1.58e5i)T + 2.59e10iT^{2} \)
13 \( 1 + (2.63e4 + 2.63e4i)T + 1.37e11iT^{2} \)
17 \( 1 - 1.46e6T + 2.01e12T^{2} \)
19 \( 1 + (-2.18e5 + 2.18e5i)T - 6.13e12iT^{2} \)
23 \( 1 + 1.19e7T + 4.14e13T^{2} \)
29 \( 1 + (-3.24e6 - 3.24e6i)T + 4.20e14iT^{2} \)
31 \( 1 - 1.71e7iT - 8.19e14T^{2} \)
37 \( 1 + (-4.84e7 + 4.84e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 2.01e8iT - 1.34e16T^{2} \)
43 \( 1 + (4.62e7 + 4.62e7i)T + 2.16e16iT^{2} \)
47 \( 1 - 3.80e8iT - 5.25e16T^{2} \)
53 \( 1 + (1.53e7 - 1.53e7i)T - 1.74e17iT^{2} \)
59 \( 1 + (-3.55e8 - 3.55e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (6.17e7 + 6.17e7i)T + 7.13e17iT^{2} \)
67 \( 1 + (8.03e8 - 8.03e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.60e8T + 3.25e18T^{2} \)
73 \( 1 - 2.16e9iT - 4.29e18T^{2} \)
79 \( 1 + 3.90e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.64e9 + 1.64e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 4.00e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.30e9T + 7.37e19T^{2} \)
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

−17.53519997891779828637895067869, −16.29620210777183983497484014754, −15.70587192114959788049798100146, −14.46070837629513117484259300303, −12.02596973659515751350732883796, −10.44925919226777122153763785977, −9.608077766421737236052962962912, −7.28226888000113081884093437071, −6.01415118226720315186218159288, −3.98822960830873583765301578144, 0.02731206609185190354809610806, 0.975853588534764412999038401212, 3.67183867057924151325487740532, 6.32873041643268503144036964799, 8.066671723542040885916415045466, 9.693057444618863063618329654047, 11.69735036604194883594042317452, 12.21539654765613457886621193457, 13.33867560965262476537831899988, 16.38125018682500172416524280185

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