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Properties

Label 2-640-80.43-c1-0-17
Degree $2$
Conductor $640$
Sign $-0.171 + 0.985i$
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  − 1.28i·3-s + (0.841 − 2.07i)5-s + (−1.13 + 1.13i)7-s + 1.35·9-s + (2.32 − 2.32i)11-s − 1.36·13-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 − 5.58i·27-s + (−2.37 − 2.37i)29-s + 0.242i·31-s + ⋯
L(s)  = 1  − 0.739i·3-s + (0.376 − 0.926i)5-s + (−0.430 + 0.430i)7-s + 0.452·9-s + (0.700 − 0.700i)11-s − 0.378·13-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 − 1.07i·27-s + (−0.440 − 0.440i)29-s + 0.0435i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.171 + 0.985i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990305 - 1.17803i\)
\(L(\frac12)\) \(\approx\) \(0.990305 - 1.17803i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.841 + 2.07i)T \)
good3 \( 1 + 1.28iT - 3T^{2} \)
7 \( 1 + (1.13 - 1.13i)T - 7iT^{2} \)
11 \( 1 + (-2.32 + 2.32i)T - 11iT^{2} \)
13 \( 1 + 1.36T + 13T^{2} \)
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 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 + 9.04T + 43T^{2} \)
47 \( 1 + (7.87 + 7.87i)T + 47iT^{2} \)
53 \( 1 - 5.80iT - 53T^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (-6.67 + 6.67i)T - 61iT^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 3.26iT - 83T^{2} \)
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

−9.976797550570743385821207153098, −9.583519632516709078886457725287, −8.506568959890215770786894958384, −7.79453831242630438227041147192, −6.66295101132503784044010015686, −5.90822318738904538272704426355, −4.91952770762883705594922981345, −3.63591260453331293383715590599, −2.14237359374816593359211641397, −0.890317407936487832552345594192, 1.84037891976303795615724343736, 3.38828178578512220070457156481, 4.09330205949848027865389545139, 5.28747811611402580297454314697, 6.53833928624663774034245696791, 7.04160403253191004132405310240, 8.189517514317924139194624854653, 9.584064037908829902041299644382, 9.866141033634750191139201468303, 10.60119065591755907700661189478

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