↑

Properties

Label 2-2e4-16.13-c13-0-21
Degree $2$
Conductor $16$
Sign $-0.584 - 0.811i$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−85.4 − 29.7i)2-s + (−660. − 660. i)3-s + (6.42e3 + 5.08e3i)4-s + (−6.98e3 + 6.98e3i)5-s + (3.68e4 + 7.61e4i)6-s − 2.94e5i·7-s + (−3.97e5 − 6.25e5i)8-s − 7.21e5i·9-s + (8.04e5 − 3.89e5i)10-s + (3.67e6 − 3.67e6i)11-s + (−8.84e5 − 7.60e6i)12-s + (−3.22e6 − 3.22e6i)13-s + (−8.75e6 + 2.51e7i)14-s + 9.22e6·15-s + (1.54e7 + 6.53e7i)16-s − 6.34e7·17-s + ⋯
L(s)  = 1  + (−0.944 − 0.328i)2-s + (−0.523 − 0.523i)3-s + (0.784 + 0.620i)4-s + (−0.199 + 0.199i)5-s + (0.322 + 0.666i)6-s − 0.945i·7-s + (−0.536 − 0.843i)8-s − 0.452i·9-s + (0.254 − 0.123i)10-s + (0.625 − 0.625i)11-s + (−0.0855 − 0.734i)12-s + (−0.185 − 0.185i)13-s + (−0.310 + 0.893i)14-s + 0.209·15-s + (0.229 + 0.973i)16-s − 0.637·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ -0.584 - 0.811i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.0734154 + 0.143384i\)
\(L(\frac12)\) \(\approx\) \(0.0734154 + 0.143384i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (85.4 + 29.7i)T \)
good3 \( 1 + (660. + 660. i)T + 1.59e6iT^{2} \)
5 \( 1 + (6.98e3 - 6.98e3i)T - 1.22e9iT^{2} \)
7 \( 1 + 2.94e5iT - 9.68e10T^{2} \)
11 \( 1 + (-3.67e6 + 3.67e6i)T - 3.45e13iT^{2} \)
13 \( 1 + (3.22e6 + 3.22e6i)T + 3.02e14iT^{2} \)
17 \( 1 + 6.34e7T + 9.90e15T^{2} \)
19 \( 1 + (4.23e7 + 4.23e7i)T + 4.20e16iT^{2} \)
23 \( 1 - 7.68e8iT - 5.04e17T^{2} \)
29 \( 1 + (2.36e9 + 2.36e9i)T + 1.02e19iT^{2} \)
31 \( 1 - 3.01e9T + 2.44e19T^{2} \)
37 \( 1 + (8.54e9 - 8.54e9i)T - 2.43e20iT^{2} \)
41 \( 1 + 2.29e10iT - 9.25e20T^{2} \)
43 \( 1 + (4.55e9 - 4.55e9i)T - 1.71e21iT^{2} \)
47 \( 1 + 1.08e11T + 5.46e21T^{2} \)
53 \( 1 + (1.84e11 - 1.84e11i)T - 2.60e22iT^{2} \)
59 \( 1 + (3.11e11 - 3.11e11i)T - 1.04e23iT^{2} \)
61 \( 1 + (1.15e11 + 1.15e11i)T + 1.61e23iT^{2} \)
67 \( 1 + (-5.19e11 - 5.19e11i)T + 5.48e23iT^{2} \)
71 \( 1 + 1.94e11iT - 1.16e24T^{2} \)
73 \( 1 + 1.97e12iT - 1.67e24T^{2} \)
79 \( 1 - 2.63e12T + 4.66e24T^{2} \)
83 \( 1 + (-2.84e12 - 2.84e12i)T + 8.87e24iT^{2} \)
89 \( 1 + 6.64e12iT - 2.19e25T^{2} \)
97 \( 1 + 7.19e11T + 6.73e25T^{2} \)
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

−15.34058844714287678210845003953, −13.36642678693142537747721902768, −11.84706983376620077801960418621, −10.90545348407827885315089455482, −9.310247643793972022054629520744, −7.55683350019672322497363662144, −6.42059271042814783250987359860, −3.55780066577609839050227701801, −1.32394580894884058199340755717, −0.094368321915907036058552881704, 2.06642047014642470239730962495, 4.88524620891015992919337440548, 6.47301002521507255786136931102, 8.319379417120123191608675524553, 9.625558498650421771005238469356, 10.99618076120090663106395703627, 12.23252837033593331434968698202, 14.63403970107408965507608818938, 15.79628089192132032966488397510, 16.71027659540618160072297824617

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