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Properties

Label 2-640-80.27-c1-0-15
Degree $2$
Conductor $640$
Sign $0.209 + 0.977i$
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  + 0.692·3-s + (0.245 − 2.22i)5-s + (0.343 + 0.343i)7-s − 2.52·9-s + (0.843 − 0.843i)11-s − 3.68i·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 − 3.82·27-s + (−4.22 − 4.22i)29-s + 8.75i·31-s + ⋯
L(s)  = 1  + 0.399·3-s + (0.109 − 0.993i)5-s + (0.129 + 0.129i)7-s − 0.840·9-s + (0.254 − 0.254i)11-s − 1.02i·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 − 0.735·27-s + (−0.785 − 0.785i)29-s + 1.57i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21466 - 0.982431i\)
\(L(\frac12)\) \(\approx\) \(1.21466 - 0.982431i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.245 + 2.22i)T \)
good3 \( 1 - 0.692T + 3T^{2} \)
7 \( 1 + (-0.343 - 0.343i)T + 7iT^{2} \)
11 \( 1 + (-0.843 + 0.843i)T - 11iT^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
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 \( 1 - 5.41iT - 37T^{2} \)
41 \( 1 + 2.54iT - 41T^{2} \)
43 \( 1 + 4.30iT - 43T^{2} \)
47 \( 1 + (-4.56 + 4.56i)T - 47iT^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + (-7.33 - 7.33i)T + 59iT^{2} \)
61 \( 1 + (-4.81 + 4.81i)T - 61iT^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
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

−10.29854069530505574282878085909, −9.284135786797210635276442793353, −8.670896470909634741736750845716, −8.033682535439326114819926110947, −6.89471240937685915089838623516, −5.53244139979421423373362061118, −5.09509459888862706849869123366, −3.62690765034955684856380565427, −2.55361058325175055912699630854, −0.836988848875452271422550818673, 1.85654756241933372921210640678, 3.08785830716292934422592811104, 3.94592655314321814737689886542, 5.44317194837534832963506951048, 6.30142295252036180559508718027, 7.37537322572528666729957353655, 7.938947551735983554849383512350, 9.346224235387218647259525760686, 9.598915650021329225881808409545, 10.97872703739951661334348248847

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