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Properties

Label 2-640-80.43-c1-0-8
Degree $2$
Conductor $640$
Sign $0.746 - 0.665i$
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  + 0.614i·3-s + (2.07 − 0.832i)5-s + (−2.83 + 2.83i)7-s + 2.62·9-s + (1.95 − 1.95i)11-s + 2.05·13-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 + 3.45i·27-s + (0.835 + 0.835i)29-s − 2.35i·31-s + ⋯
L(s)  = 1  + 0.354i·3-s + (0.928 − 0.372i)5-s + (−1.07 + 1.07i)7-s + 0.874·9-s + (0.590 − 0.590i)11-s + 0.569·13-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 + 0.664i·27-s + (0.155 + 0.155i)29-s − 0.423i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.746 - 0.665i$
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: \((2,\ 640,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60583 + 0.611954i\)
\(L(\frac12)\) \(\approx\) \(1.60583 + 0.611954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 + 0.832i)T \)
good3 \( 1 - 0.614iT - 3T^{2} \)
7 \( 1 + (2.83 - 2.83i)T - 7iT^{2} \)
11 \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
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 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 0.849T + 43T^{2} \)
47 \( 1 + (2.72 + 2.72i)T + 47iT^{2} \)
53 \( 1 + 5.17iT - 53T^{2} \)
59 \( 1 + (-4.16 - 4.16i)T + 59iT^{2} \)
61 \( 1 + (5.55 - 5.55i)T - 61iT^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 + 4.39i)T - 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
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

−10.54517754031252440479227874254, −9.535075982559444257116209539519, −9.219500782015823912052025679417, −8.391092827102797896065237332210, −6.78238777788166882611726947411, −6.17644355030787386258908981653, −5.33886050191975993286903002555, −4.08719123567622421787600017211, −2.93701160924099676544638366034, −1.52882454394347647395753533362, 1.09772487669820153631363418858, 2.54640422642682779580744621748, 3.83231208088717556263034619010, 4.86757304510667032750558558395, 6.54849541028351100477958884211, 6.62132552160853639280945625058, 7.50261533446680152252724296516, 9.042959580972371259648045098282, 9.646112278291118945555987552767, 10.38911869991998257757957313399

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