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Properties

Label 2-640-80.3-c1-0-18
Degree $2$
Conductor $640$
Sign $0.754 + 0.656i$
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  + 2.96·3-s + (0.177 − 2.22i)5-s + (−0.115 + 0.115i)7-s + 5.79·9-s + (−2.95 − 2.95i)11-s − 1.55i·13-s + (0.525 − 6.61i)15-s + (0.299 − 0.299i)17-s + (2.26 + 2.26i)19-s + (−0.341 + 0.341i)21-s + (4.14 + 4.14i)23-s + (−4.93 − 0.790i)25-s + 8.28·27-s + (−0.289 + 0.289i)29-s + 4.18i·31-s + ⋯
L(s)  = 1  + 1.71·3-s + (0.0793 − 0.996i)5-s + (−0.0435 + 0.0435i)7-s + 1.93·9-s + (−0.892 − 0.892i)11-s − 0.432i·13-s + (0.135 − 1.70i)15-s + (0.0726 − 0.0726i)17-s + (0.519 + 0.519i)19-s + (−0.0744 + 0.0744i)21-s + (0.864 + 0.864i)23-s + (−0.987 − 0.158i)25-s + 1.59·27-s + (−0.0537 + 0.0537i)29-s + 0.751i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.754 + 0.656i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.754 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40230 - 0.899418i\)
\(L(\frac12)\) \(\approx\) \(2.40230 - 0.899418i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.177 + 2.22i)T \)
good3 \( 1 - 2.96T + 3T^{2} \)
7 \( 1 + (0.115 - 0.115i)T - 7iT^{2} \)
11 \( 1 + (2.95 + 2.95i)T + 11iT^{2} \)
13 \( 1 + 1.55iT - 13T^{2} \)
17 \( 1 + (-0.299 + 0.299i)T - 17iT^{2} \)
19 \( 1 + (-2.26 - 2.26i)T + 19iT^{2} \)
23 \( 1 + (-4.14 - 4.14i)T + 23iT^{2} \)
29 \( 1 + (0.289 - 0.289i)T - 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 - 1.63iT - 37T^{2} \)
41 \( 1 + 7.61iT - 41T^{2} \)
43 \( 1 - 6.72iT - 43T^{2} \)
47 \( 1 + (-4.38 - 4.38i)T + 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (1.63 - 1.63i)T - 59iT^{2} \)
61 \( 1 + (-1.23 - 1.23i)T + 61iT^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (1.12 - 1.12i)T - 73iT^{2} \)
79 \( 1 - 3.62T + 79T^{2} \)
83 \( 1 + 1.62T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-9.69 + 9.69i)T - 97iT^{2} \)
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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

−10.15998961319836537445861513309, −9.351897106181089819495376480658, −8.724083483559897141105692229903, −8.005710098940204849545783167655, −7.42725699773483851910902294653, −5.80712857593509888846406618859, −4.82748874537811179836790567093, −3.56116130838141409390513279695, −2.78905566470379776203212923321, −1.34994820448317692839902697230, 2.09748460386978569731765945915, 2.79205486307647821339652754029, 3.77402399913411434632116479633, 4.93320374083286539778276715812, 6.57843016087419175571629062134, 7.35167372423813612944745696319, 7.954503204362120078043085514125, 8.983671427862099999626651311132, 9.740968539531446635825847248624, 10.38025623041091199297865487323

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