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Properties

Label 2-640-80.27-c1-0-1
Degree $2$
Conductor $640$
Sign $-0.878 - 0.477i$
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  − 0.614·3-s + (−0.832 + 2.07i)5-s + (2.83 + 2.83i)7-s − 2.62·9-s + (−1.95 + 1.95i)11-s − 2.05i·13-s + (0.511 − 1.27i)15-s + (−4.06 − 4.06i)17-s + (0.683 − 0.683i)19-s + (−1.74 − 1.74i)21-s + (−4.95 + 4.95i)23-s + (−3.61 − 3.45i)25-s + 3.45·27-s + (−0.835 − 0.835i)29-s + 2.35i·31-s + ⋯
L(s)  = 1  − 0.354·3-s + (−0.372 + 0.928i)5-s + (1.07 + 1.07i)7-s − 0.874·9-s + (−0.590 + 0.590i)11-s − 0.569i·13-s + (0.132 − 0.329i)15-s + (−0.986 − 0.986i)17-s + (0.156 − 0.156i)19-s + (−0.380 − 0.380i)21-s + (−1.03 + 1.03i)23-s + (−0.723 − 0.690i)25-s + 0.664·27-s + (−0.155 − 0.155i)29-s + 0.423i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.878 - 0.477i$
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: \((2,\ 640,\ (\ :1/2),\ -0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171138 + 0.673061i\)
\(L(\frac12)\) \(\approx\) \(0.171138 + 0.673061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.832 - 2.07i)T \)
good3 \( 1 + 0.614T + 3T^{2} \)
7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (4.06 + 4.06i)T + 17iT^{2} \)
19 \( 1 + (-0.683 + 0.683i)T - 19iT^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + (0.835 + 0.835i)T + 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 0.849iT - 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 + 5.17T + 53T^{2} \)
59 \( 1 + (-4.16 - 4.16i)T + 59iT^{2} \)
61 \( 1 + (5.55 - 5.55i)T - 61iT^{2} \)
67 \( 1 - 1.73iT - 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (4.39 + 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−11.17739383562988267064836981253, −10.23575523476755643303106896317, −9.169189519159133401636515045050, −8.167727161753928275611573679281, −7.56802769908850223924657462058, −6.38419072683566774858037825424, −5.46348277351508512623088931584, −4.68973612594959486978633499124, −3.06233159973919323636833072583, −2.19153746471552192857942832479, 0.37404651669790549236409014261, 1.94534089260288521522386119641, 3.83670027212164678937935964299, 4.60377779569130164930426754941, 5.51718001136990744628867277875, 6.56180949448322427233064977212, 7.911736562117787295014925350455, 8.263250186237577092421459156819, 9.164840248708103389391127939050, 10.55984154139481004478092836817

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