↑

Properties

Label 2-2e5-32.11-c2-0-6
Degree $2$
Conductor $32$
Sign $-0.463 + 0.885i$
Analytic cond. $0.871936$
Root an. cond. $0.933775$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.658 − 1.88i)2-s + (−1.31 − 3.18i)3-s + (−3.13 + 2.48i)4-s + (−0.659 + 1.59i)5-s + (−5.14 + 4.58i)6-s + (9.54 − 9.54i)7-s + (6.75 + 4.27i)8-s + (−2.03 + 2.03i)9-s + (3.44 + 0.197i)10-s + (−3.96 + 9.57i)11-s + (12.0 + 6.69i)12-s + (1.91 + 4.63i)13-s + (−24.3 − 11.7i)14-s + 5.93·15-s + (3.62 − 15.5i)16-s + 15.3i·17-s + ⋯
L(s)  = 1  + (−0.329 − 0.944i)2-s + (−0.439 − 1.06i)3-s + (−0.783 + 0.621i)4-s + (−0.131 + 0.318i)5-s + (−0.857 + 0.764i)6-s + (1.36 − 1.36i)7-s + (0.844 + 0.534i)8-s + (−0.225 + 0.225i)9-s + (0.344 + 0.0197i)10-s + (−0.360 + 0.870i)11-s + (1.00 + 0.557i)12-s + (0.147 + 0.356i)13-s + (−1.73 − 0.838i)14-s + 0.395·15-s + (0.226 − 0.973i)16-s + 0.900i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.885i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.463 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.463 + 0.885i$
Analytic conductor: \(0.871936\)
Root analytic conductor: \(0.933775\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :1),\ -0.463 + 0.885i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.400726 - 0.662216i\)
\(L(\frac12)\) \(\approx\) \(0.400726 - 0.662216i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.658 + 1.88i)T \)
good3 \( 1 + (1.31 + 3.18i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.659 - 1.59i)T + (-17.6 - 17.6i)T^{2} \)
7 \( 1 + (-9.54 + 9.54i)T - 49iT^{2} \)
11 \( 1 + (3.96 - 9.57i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.91 - 4.63i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 15.3iT - 289T^{2} \)
19 \( 1 + (-0.827 + 0.342i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (12.9 + 12.9i)T + 529iT^{2} \)
29 \( 1 + (-23.7 + 9.85i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 25.1iT - 961T^{2} \)
37 \( 1 + (-13.6 + 32.8i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (32.9 - 32.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (17.9 - 43.3i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 20.1T + 2.20e3T^{2} \)
53 \( 1 + (-35.0 - 14.5i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (60.6 + 25.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (27.9 - 11.5i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-1.13 - 2.73i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (45.6 - 45.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (29.1 - 29.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.27T + 6.24e3T^{2} \)
83 \( 1 + (-56.7 + 23.5i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (44.5 + 44.5i)T + 7.92e3iT^{2} \)
97 \( 1 + 106.T + 9.40e3T^{2} \)
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

−16.90338382208349834523431245850, −14.58737183327369982970113059412, −13.46753172447091600263419435340, −12.36176281144796467704195731378, −11.22979149010193256606224144299, −10.28015864324980150023279802694, −8.114397345034232444306682363849, −7.11533745485830620515438711844, −4.42212235834009396529665018768, −1.50398682081496164812246883691, 4.81080622760196026239547663029, 5.64459262471308738477329107127, 8.059149180418712813233493162266, 9.041426195509084103492468733699, 10.54188745334595960669493894890, 11.80884881592401359920472129607, 13.80945490559753575129921145925, 15.14099092145470307234447853821, 15.75672535532746314142054626402, 16.72766222435310141241781493319

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