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Properties

Label 2-640-80.3-c1-0-17
Degree $2$
Conductor $640$
Sign $-0.135 + 0.990i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.135 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846895 - 0.970179i\)
\(L(\frac12)\) \(\approx\) \(0.846895 - 0.970179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.38324533726620545960640645184, −9.270160112430561092267469398307, −8.406714301050292923082922304623, −7.913995301843793608257920075422, −7.13021838672878654003762136824, −5.60376770765734605105010943757, −4.72313963189504555776002663356, −3.51184051401416159189289317636, −2.68726873822622307637762247325, −0.63499248154432419470745605617, 1.97514335423496007660304446139, 3.14997914396467321975208181854, 4.07176113729993996002571629699, 5.27051449636712729208561127881, 6.41811716084742925623892320050, 7.63158644804705180272611899153, 8.175478968793331716691692680925, 8.764405669204909395771162951145, 10.06327812346871901585595817493, 10.69297034077292944110361064149

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