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Properties

Label 2-640-40.3-c1-0-6
Degree $2$
Conductor $640$
Sign $0.229 - 0.973i$
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 + 2i)3-s + (−2 − i)5-s + (2 + 2i)7-s + 5i·9-s + 4·11-s + (3 − 3i)13-s + (−2 − 6i)15-s + (−3 + 3i)17-s + 8i·21-s + (−6 + 6i)23-s + (3 + 4i)25-s + (−4 + 4i)27-s − 2·29-s − 4i·31-s + (8 + 8i)33-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)3-s + (−0.894 − 0.447i)5-s + (0.755 + 0.755i)7-s + 1.66i·9-s + 1.20·11-s + (0.832 − 0.832i)13-s + (−0.516 − 1.54i)15-s + (−0.727 + 0.727i)17-s + 1.74i·21-s + (−1.25 + 1.25i)23-s + (0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s − 0.371·29-s − 0.718i·31-s + (1.39 + 1.39i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64644 + 1.30302i\)
\(L(\frac12)\) \(\approx\) \(1.64644 + 1.30302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
good3 \( 1 + (-2 - 2i)T + 3iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (6 - 6i)T - 23iT^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (6 + 6i)T + 43iT^{2} \)
47 \( 1 + (-6 - 6i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 + (-6 + 6i)T - 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-6 - 6i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-11 + 11i)T - 97iT^{2} \)
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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

−10.78000155244330814722909761744, −9.595837863586471341676557871521, −8.973402191308713849250181184958, −8.277194263967601432643402914382, −7.80462613316474300730407221303, −6.11135016692257192349242371734, −4.93993805467126066029941428681, −3.97752863634768898288480899700, −3.46382502047251490782923274322, −1.85686263916154129754932742731, 1.16603695945558309403988358106, 2.37312902068764667396529395229, 3.74578683485653843943465030451, 4.34712349221846685579644102941, 6.47946745621833698467095916299, 6.93457328775907997463927279719, 7.76502993630664451722651564132, 8.507285587563172611323430280318, 9.106066377149385055643637010456, 10.49953090973320678114938046979

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