↑

Properties

Label 2-2e4-16.3-c8-0-10
Degree $2$
Conductor $16$
Sign $0.979 - 0.201i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.41 + 15.0i)2-s + (63.2 − 63.2i)3-s + (−197. + 163. i)4-s + (825. − 825. i)5-s + (1.29e3 + 609. i)6-s + 711.·7-s + (−3.52e3 − 2.08e3i)8-s − 1.44e3i·9-s + (1.69e4 + 7.96e3i)10-s + (7.26e3 + 7.26e3i)11-s + (−2.17e3 + 2.27e4i)12-s + (1.47e4 + 1.47e4i)13-s + (3.85e3 + 1.07e4i)14-s − 1.04e5i·15-s + (1.23e4 − 6.43e4i)16-s − 1.18e5·17-s + ⋯
L(s)  = 1  + (0.338 + 0.941i)2-s + (0.780 − 0.780i)3-s + (−0.771 + 0.636i)4-s + (1.32 − 1.32i)5-s + (0.999 + 0.470i)6-s + 0.296·7-s + (−0.860 − 0.510i)8-s − 0.219i·9-s + (1.69 + 0.796i)10-s + (0.495 + 0.495i)11-s + (−0.104 + 1.09i)12-s + (0.516 + 0.516i)13-s + (0.100 + 0.278i)14-s − 2.06i·15-s + (0.188 − 0.981i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.979 - 0.201i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ 0.979 - 0.201i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.62104 + 0.266435i\)
\(L(\frac12)\) \(\approx\) \(2.62104 + 0.266435i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.41 - 15.0i)T \)
good3 \( 1 + (-63.2 + 63.2i)T - 6.56e3iT^{2} \)
5 \( 1 + (-825. + 825. i)T - 3.90e5iT^{2} \)
7 \( 1 - 711.T + 5.76e6T^{2} \)
11 \( 1 + (-7.26e3 - 7.26e3i)T + 2.14e8iT^{2} \)
13 \( 1 + (-1.47e4 - 1.47e4i)T + 8.15e8iT^{2} \)
17 \( 1 + 1.18e5T + 6.97e9T^{2} \)
19 \( 1 + (-4.19e4 + 4.19e4i)T - 1.69e10iT^{2} \)
23 \( 1 + 2.55e5T + 7.83e10T^{2} \)
29 \( 1 + (3.51e5 + 3.51e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 1.79e6iT - 8.52e11T^{2} \)
37 \( 1 + (1.67e6 - 1.67e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 8.12e5iT - 7.98e12T^{2} \)
43 \( 1 + (-1.19e6 - 1.19e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 4.04e6iT - 2.38e13T^{2} \)
53 \( 1 + (-3.74e6 + 3.74e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (6.43e6 + 6.43e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (5.59e6 + 5.59e6i)T + 1.91e14iT^{2} \)
67 \( 1 + (-9.83e6 + 9.83e6i)T - 4.06e14iT^{2} \)
71 \( 1 - 1.65e7T + 6.45e14T^{2} \)
73 \( 1 + 2.55e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.22e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.67e7 + 1.67e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 1.07e7iT - 3.93e15T^{2} \)
97 \( 1 - 9.36e6T + 7.83e15T^{2} \)
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

−17.30235628203037868481534281177, −15.98593648621232014294049517516, −14.11947420884984651315691712842, −13.52633241072637661272783254228, −12.49267216115990090189630683191, −9.280326514068234185112085327258, −8.398254944862746178955088785407, −6.56613930671177664844834773985, −4.81544150123498362516599105056, −1.73256000370276193906769956166, 2.24213511524782576738699487075, 3.66858973427480742081847192043, 6.01364147924902771696906613755, 9.031106144247262065859953139558, 10.11952693450775123589974104113, 11.16786477706144667352727776134, 13.46594661220178026282302423183, 14.28080426274239330279545999165, 15.21476743278171433767334876016, 17.66254279246409869821357805704

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