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Properties

Label 2-640-80.27-c1-0-3
Degree $2$
Conductor $640$
Sign $-0.510 - 0.859i$
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  − 0.496·3-s + (2.00 + 0.987i)5-s + (−1.55 − 1.55i)7-s − 2.75·9-s + (−4.19 + 4.19i)11-s + 5.09i·13-s + (−0.996 − 0.490i)15-s + (0.213 + 0.213i)17-s + (0.844 − 0.844i)19-s + (0.771 + 0.771i)21-s + (−1.70 + 1.70i)23-s + (3.05 + 3.96i)25-s + 2.85·27-s + (−2.24 − 2.24i)29-s − 0.818i·31-s + ⋯
L(s)  = 1  − 0.286·3-s + (0.897 + 0.441i)5-s + (−0.587 − 0.587i)7-s − 0.917·9-s + (−1.26 + 1.26i)11-s + 1.41i·13-s + (−0.257 − 0.126i)15-s + (0.0517 + 0.0517i)17-s + (0.193 − 0.193i)19-s + (0.168 + 0.168i)21-s + (−0.356 + 0.356i)23-s + (0.610 + 0.792i)25-s + 0.549·27-s + (−0.417 − 0.417i)29-s − 0.146i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.510 - 0.859i$
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: \((2,\ 640,\ (\ :1/2),\ -0.510 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402379 + 0.706844i\)
\(L(\frac12)\) \(\approx\) \(0.402379 + 0.706844i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.00 - 0.987i)T \)
good3 \( 1 + 0.496T + 3T^{2} \)
7 \( 1 + (1.55 + 1.55i)T + 7iT^{2} \)
11 \( 1 + (4.19 - 4.19i)T - 11iT^{2} \)
13 \( 1 - 5.09iT - 13T^{2} \)
17 \( 1 + (-0.213 - 0.213i)T + 17iT^{2} \)
19 \( 1 + (-0.844 + 0.844i)T - 19iT^{2} \)
23 \( 1 + (1.70 - 1.70i)T - 23iT^{2} \)
29 \( 1 + (2.24 + 2.24i)T + 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 - 5.12iT - 37T^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 - 4.49iT - 43T^{2} \)
47 \( 1 + (4.29 - 4.29i)T - 47iT^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + (7.65 + 7.65i)T + 59iT^{2} \)
61 \( 1 + (-1.90 + 1.90i)T - 61iT^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 - 2.70i)T + 73iT^{2} \)
79 \( 1 - 8.32T + 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 + 4.25T + 89T^{2} \)
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

−10.83854711475956689630731676089, −9.782883953386508603852269637496, −9.594606425593410841542941901863, −8.193101561564598426987474599987, −7.11688957558974399288785258277, −6.46286237480856668395471015239, −5.47482193366919656412494035462, −4.49694330440363581895641527555, −3.02204307014914804749768796897, −1.95986041832856258122224335548, 0.42197947806832405779336191530, 2.49952968456717480316842317055, 3.28029329201179198923003865524, 5.34765831154805846964818264178, 5.51911339853702316497309378016, 6.32920977580182493145502081351, 7.889697777747404272959642849647, 8.568023665853624888848368670201, 9.364324882477937571044820242875, 10.47540110290967361728336251893

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