↑

Properties

Label 2-160-40.19-c4-0-17
Degree $2$
Conductor $160$
Sign $-0.345 + 0.938i$
Analytic cond. $16.5391$
Root an. cond. $4.06684$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.05i·3-s + (−4.74 − 24.5i)5-s + 65.1·7-s − 0.965·9-s + 220.·11-s − 75.1·13-s + (−222. + 42.9i)15-s − 341. i·17-s + 59.8·19-s − 590. i·21-s − 449.·23-s + (−579. + 233. i)25-s − 724. i·27-s + 977. i·29-s − 89.3i·31-s + ⋯
L(s)  = 1  − 1.00i·3-s + (−0.189 − 0.981i)5-s + 1.33·7-s − 0.0119·9-s + 1.81·11-s − 0.444·13-s + (−0.987 + 0.191i)15-s − 1.18i·17-s + 0.165·19-s − 1.33i·21-s − 0.848·23-s + (−0.927 + 0.372i)25-s − 0.993i·27-s + 1.16i·29-s − 0.0930i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.345 + 0.938i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ -0.345 + 0.938i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.25303 - 1.79620i\)
\(L(\frac12)\) \(\approx\) \(1.25303 - 1.79620i\)
\(L(3)\) 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 \( 1 + (4.74 + 24.5i)T \)
good3 \( 1 + 9.05iT - 81T^{2} \)
7 \( 1 - 65.1T + 2.40e3T^{2} \)
11 \( 1 - 220.T + 1.46e4T^{2} \)
13 \( 1 + 75.1T + 2.85e4T^{2} \)
17 \( 1 + 341. iT - 8.35e4T^{2} \)
19 \( 1 - 59.8T + 1.30e5T^{2} \)
23 \( 1 + 449.T + 2.79e5T^{2} \)
29 \( 1 - 977. iT - 7.07e5T^{2} \)
31 \( 1 + 89.3iT - 9.23e5T^{2} \)
37 \( 1 + 1.48e3T + 1.87e6T^{2} \)
41 \( 1 - 1.55e3T + 2.82e6T^{2} \)
43 \( 1 - 245. iT - 3.41e6T^{2} \)
47 \( 1 + 180.T + 4.87e6T^{2} \)
53 \( 1 + 1.24e3T + 7.89e6T^{2} \)
59 \( 1 - 4.05e3T + 1.21e7T^{2} \)
61 \( 1 + 3.52e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.15e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.61e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.40e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.92e3iT - 3.89e7T^{2} \)
83 \( 1 - 7.72e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.58e3T + 6.27e7T^{2} \)
97 \( 1 - 2.35e3iT - 8.85e7T^{2} \)
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

−11.96409141109020663083231532299, −11.38323464294206876182760576826, −9.611656793132337134041806913358, −8.646232718404661463420982128475, −7.68576354622029040844503071167, −6.75872261031382185871641627364, −5.22344183722650459729982530236, −4.14940778447766177556520834056, −1.81218388774105864751531978494, −0.959677945508520984684914031354, 1.77063943422811775685127513364, 3.74207696618920689716302825567, 4.44326426306121711087363683282, 6.02054833407971857167847747032, 7.27505506578439355350797623365, 8.456144482677829230119699963696, 9.647759915907638705395035805085, 10.51854921037599929382100348941, 11.38131623897420597271944748200, 12.11368985462837262840458749872

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