↑

Properties

Label 2-320-80.3-c1-0-3
Degree $2$
Conductor $320$
Sign $0.962 - 0.272i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.28·3-s + (2.07 + 0.841i)5-s + (1.13 − 1.13i)7-s − 1.35·9-s + (2.32 + 2.32i)11-s + 1.36i·13-s + (−2.65 − 1.07i)15-s + (5.25 − 5.25i)17-s + (3.69 + 3.69i)19-s + (−1.46 + 1.46i)21-s + (0.911 + 0.911i)23-s + (3.58 + 3.48i)25-s + 5.58·27-s + (−2.37 + 2.37i)29-s + 0.242i·31-s + ⋯
L(s)  = 1  − 0.739·3-s + (0.926 + 0.376i)5-s + (0.430 − 0.430i)7-s − 0.452·9-s + (0.700 + 0.700i)11-s + 0.378i·13-s + (−0.685 − 0.278i)15-s + (1.27 − 1.27i)17-s + (0.848 + 0.848i)19-s + (−0.318 + 0.318i)21-s + (0.189 + 0.189i)23-s + (0.716 + 0.697i)25-s + 1.07·27-s + (−0.440 + 0.440i)29-s + 0.0435i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.962 - 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25841 + 0.174663i\)
\(L(\frac12)\) \(\approx\) \(1.25841 + 0.174663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.07 - 0.841i)T \)
good3 \( 1 + 1.28T + 3T^{2} \)
7 \( 1 + (-1.13 + 1.13i)T - 7iT^{2} \)
11 \( 1 + (-2.32 - 2.32i)T + 11iT^{2} \)
13 \( 1 - 1.36iT - 13T^{2} \)
17 \( 1 + (-5.25 + 5.25i)T - 17iT^{2} \)
19 \( 1 + (-3.69 - 3.69i)T + 19iT^{2} \)
23 \( 1 + (-0.911 - 0.911i)T + 23iT^{2} \)
29 \( 1 + (2.37 - 2.37i)T - 29iT^{2} \)
31 \( 1 - 0.242iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + 2.66iT - 41T^{2} \)
43 \( 1 + 9.04iT - 43T^{2} \)
47 \( 1 + (7.87 + 7.87i)T + 47iT^{2} \)
53 \( 1 + 5.80T + 53T^{2} \)
59 \( 1 + (5.91 - 5.91i)T - 59iT^{2} \)
61 \( 1 + (6.67 + 6.67i)T + 61iT^{2} \)
67 \( 1 + 4.54iT - 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 3.26T + 83T^{2} \)
89 \( 1 - 9.77T + 89T^{2} \)
97 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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

−11.74215334666037977874579563830, −10.75336985169969215692048619681, −9.891226594728909539389341249289, −9.135325010730621799959460976302, −7.62180566792580932732079566669, −6.76261375172642875677543275294, −5.68227539867006571083885705140, −4.91835643461468848556835950144, −3.24646111281110138870280094741, −1.49558833181212013160910252157, 1.28371921894085986235664820501, 3.07874504779616669927586336970, 4.84968745250086011243012732495, 5.78300723755754378114199750072, 6.26079086015039815180791440833, 7.912691568097992239680289705413, 8.843609263779571377324739308962, 9.735636108071792190883180193342, 10.81436782675659931362141151241, 11.55048233310251982107697308956

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