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Properties

Label 2-640-80.43-c1-0-7
Degree $2$
Conductor $640$
Sign $0.626 - 0.779i$
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  + 0.692i·3-s + (−2.22 + 0.245i)5-s + (0.343 − 0.343i)7-s + 2.52·9-s + (0.843 − 0.843i)11-s + 3.68·13-s + (−0.169 − 1.53i)15-s + (0.412 − 0.412i)17-s + (−5.37 + 5.37i)19-s + (0.238 + 0.238i)21-s + (3.08 + 3.08i)23-s + (4.87 − 1.09i)25-s + 3.82i·27-s + (4.22 + 4.22i)29-s + 8.75i·31-s + ⋯
L(s)  = 1  + 0.399i·3-s + (−0.993 + 0.109i)5-s + (0.129 − 0.129i)7-s + 0.840·9-s + (0.254 − 0.254i)11-s + 1.02·13-s + (−0.0438 − 0.397i)15-s + (0.0999 − 0.0999i)17-s + (−1.23 + 1.23i)19-s + (0.0519 + 0.0519i)21-s + (0.643 + 0.643i)23-s + (0.975 − 0.218i)25-s + 0.735i·27-s + (0.785 + 0.785i)29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.626 - 0.779i$
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: \((2,\ 640,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23285 + 0.591084i\)
\(L(\frac12)\) \(\approx\) \(1.23285 + 0.591084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.245i)T \)
good3 \( 1 - 0.692iT - 3T^{2} \)
7 \( 1 + (-0.343 + 0.343i)T - 7iT^{2} \)
11 \( 1 + (-0.843 + 0.843i)T - 11iT^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (-0.412 + 0.412i)T - 17iT^{2} \)
19 \( 1 + (5.37 - 5.37i)T - 19iT^{2} \)
23 \( 1 + (-3.08 - 3.08i)T + 23iT^{2} \)
29 \( 1 + (-4.22 - 4.22i)T + 29iT^{2} \)
31 \( 1 - 8.75iT - 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 2.54iT - 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (4.56 + 4.56i)T + 47iT^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + (7.33 + 7.33i)T + 59iT^{2} \)
61 \( 1 + (-4.81 + 4.81i)T - 61iT^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + (6.87 - 6.87i)T - 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 7.15iT - 83T^{2} \)
89 \( 1 - 1.10T + 89T^{2} \)
97 \( 1 + (-7.15 + 7.15i)T - 97iT^{2} \)
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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

−10.78161983888932376724049121154, −9.981672215135658184262781250064, −8.802264385663587609089018413603, −8.197849430318724364050464532115, −7.16429068918066227573245794268, −6.34561572788761351766266081845, −4.98477091198117871099795542596, −4.02322895381993157536077487015, −3.35012312001084082444447448895, −1.35306319744508917035148097596, 0.905766668427814484239508247854, 2.53154120753009476581167194733, 4.06870361376216914902194604815, 4.58014213452920086901437775759, 6.19175093993133646171613934723, 6.91118067057403667480468942215, 7.85823596563319881268027843768, 8.551972760255469151750036637606, 9.483447369070089800744587418184, 10.66659223200942502055456979873

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