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Properties

Label 2-640-16.13-c3-0-20
Degree $2$
Conductor $640$
Sign $-0.537 + 0.843i$
Analytic cond. $37.7612$
Root an. cond. $6.14501$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.39 − 6.39i)3-s + (3.53 − 3.53i)5-s + 30.7i·7-s + 54.7i·9-s + (−37.5 + 37.5i)11-s + (−22.1 − 22.1i)13-s − 45.2·15-s − 27.4·17-s + (46.7 + 46.7i)19-s + (196. − 196. i)21-s − 21.0i·23-s − 25.0i·25-s + (177. − 177. i)27-s + (−61.1 − 61.1i)29-s − 75.2·31-s + ⋯
L(s)  = 1  + (−1.23 − 1.23i)3-s + (0.316 − 0.316i)5-s + 1.66i·7-s + 2.02i·9-s + (−1.02 + 1.02i)11-s + (−0.472 − 0.472i)13-s − 0.778·15-s − 0.392·17-s + (0.564 + 0.564i)19-s + (2.04 − 2.04i)21-s − 0.190i·23-s − 0.200i·25-s + (1.26 − 1.26i)27-s + (−0.391 − 0.391i)29-s − 0.435·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.537 + 0.843i$
Analytic conductor: \(37.7612\)
Root analytic conductor: \(6.14501\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :3/2),\ -0.537 + 0.843i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5403356437\)
\(L(\frac12)\) \(\approx\) \(0.5403356437\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.53 + 3.53i)T \)
good3 \( 1 + (6.39 + 6.39i)T + 27iT^{2} \)
7 \( 1 - 30.7iT - 343T^{2} \)
11 \( 1 + (37.5 - 37.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (22.1 + 22.1i)T + 2.19e3iT^{2} \)
17 \( 1 + 27.4T + 4.91e3T^{2} \)
19 \( 1 + (-46.7 - 46.7i)T + 6.85e3iT^{2} \)
23 \( 1 + 21.0iT - 1.21e4T^{2} \)
29 \( 1 + (61.1 + 61.1i)T + 2.43e4iT^{2} \)
31 \( 1 + 75.2T + 2.97e4T^{2} \)
37 \( 1 + (-236. + 236. i)T - 5.06e4iT^{2} \)
41 \( 1 + 317. iT - 6.89e4T^{2} \)
43 \( 1 + (139. - 139. i)T - 7.95e4iT^{2} \)
47 \( 1 - 428.T + 1.03e5T^{2} \)
53 \( 1 + (-140. + 140. i)T - 1.48e5iT^{2} \)
59 \( 1 + (136. - 136. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-65.1 - 65.1i)T + 2.26e5iT^{2} \)
67 \( 1 + (692. + 692. i)T + 3.00e5iT^{2} \)
71 \( 1 + 647. iT - 3.57e5T^{2} \)
73 \( 1 - 502. iT - 3.89e5T^{2} \)
79 \( 1 + 251.T + 4.93e5T^{2} \)
83 \( 1 + (482. + 482. i)T + 5.71e5iT^{2} \)
89 \( 1 - 887. iT - 7.04e5T^{2} \)
97 \( 1 - 1.12e3T + 9.12e5T^{2} \)
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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

−9.967201855770174834827822034015, −8.968200922210947537838052658994, −7.84720771939870630182749659107, −7.23747568999686996089357545545, −5.97276050248581711356503377171, −5.58811324173664456433287571898, −4.81278474469668392839652504164, −2.50637039341440762542845975464, −1.85918476455757368174766075221, −0.24119516374466615839385233326, 0.849954560365351226822390645723, 3.08173197322981514159991224824, 4.13951384996627054355001577606, 4.91275071014631909246455912555, 5.80480495711979929786592830006, 6.76621864220085584245219873196, 7.63928560688977448245906064639, 9.063252027576408905918774819350, 10.03409739433253924533094911645, 10.42542726800305209891130330859

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