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Properties

Label 2-640-16.5-c1-0-4
Degree $2$
Conductor $640$
Sign $0.163 - 0.986i$
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  + (0.720 − 0.720i)3-s + (−0.707 − 0.707i)5-s + 4.02i·7-s + 1.96i·9-s + (0.646 + 0.646i)11-s + (−4.91 + 4.91i)13-s − 1.01·15-s − 2.70·17-s + (0.438 − 0.438i)19-s + (2.90 + 2.90i)21-s − 3.60i·23-s + 1.00i·25-s + (3.57 + 3.57i)27-s + (−2.00 + 2.00i)29-s + 4.30·31-s + ⋯
L(s)  = 1  + (0.416 − 0.416i)3-s + (−0.316 − 0.316i)5-s + 1.52i·7-s + 0.653i·9-s + (0.195 + 0.195i)11-s + (−1.36 + 1.36i)13-s − 0.263·15-s − 0.656·17-s + (0.100 − 0.100i)19-s + (0.633 + 0.633i)21-s − 0.750i·23-s + 0.200i·25-s + (0.688 + 0.688i)27-s + (−0.373 + 0.373i)29-s + 0.774·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965074 + 0.818598i\)
\(L(\frac12)\) \(\approx\) \(0.965074 + 0.818598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.720 + 0.720i)T - 3iT^{2} \)
7 \( 1 - 4.02iT - 7T^{2} \)
11 \( 1 + (-0.646 - 0.646i)T + 11iT^{2} \)
13 \( 1 + (4.91 - 4.91i)T - 13iT^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + (-0.438 + 0.438i)T - 19iT^{2} \)
23 \( 1 + 3.60iT - 23T^{2} \)
29 \( 1 + (2.00 - 2.00i)T - 29iT^{2} \)
31 \( 1 - 4.30T + 31T^{2} \)
37 \( 1 + (-0.743 - 0.743i)T + 37iT^{2} \)
41 \( 1 - 0.603iT - 41T^{2} \)
43 \( 1 + (-5.03 - 5.03i)T + 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (4.07 + 4.07i)T + 53iT^{2} \)
59 \( 1 + (1.22 + 1.22i)T + 59iT^{2} \)
61 \( 1 + (-6.98 + 6.98i)T - 61iT^{2} \)
67 \( 1 + (5.24 - 5.24i)T - 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 1.30iT - 73T^{2} \)
79 \( 1 - 0.611T + 79T^{2} \)
83 \( 1 + (1.29 - 1.29i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−10.90154172675631647239889023120, −9.571666294530134483966528027480, −9.005597590502028678954981105636, −8.247954061771273039319427391468, −7.30392112278013418804713150423, −6.40840734337038430272632999791, −5.14636118904516430302900811541, −4.43306628602676412902943478842, −2.65163688374248633813614099861, −2.00323305257107954349437330457, 0.64000854838136507133344940303, 2.77840514930882530567667365528, 3.76115384138428170783012521035, 4.51432414719872237127298040540, 5.86641372493034028575515154285, 7.15048453363496393248619540237, 7.54445009136563694615461447529, 8.652371152139359367382411821835, 9.755444480216955132442835786280, 10.23832399634468134983030122507

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