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Properties

Label 2-640-16.13-c3-0-20
Degree 22
Conductor 640640
Sign −0.537+0.843i-0.537 + 0.843i
Analytic cond. 37.761237.7612
Root an. cond. 6.145016.14501
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(640s/2 ΓC(s) L(s)=((−0.537+0.843i) Λ‾(4−s)\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}
Λ(s)=(640s/2 ΓC(s+3/2) L(s)=((−0.537+0.843i) Λ‾(1−s)\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: 22
Conductor: 640640    =    27⋅52^{7} \cdot 5
Sign: −0.537+0.843i-0.537 + 0.843i
Analytic conductor: 37.761237.7612
Root analytic conductor: 6.145016.14501
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ640(161,⋅)\chi_{640} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 640, ( :3/2), −0.537+0.843i)(2,\ 640,\ (\ :3/2),\ -0.537 + 0.843i)

Particular Values

L(2)L(2) ≈\approx 0.54033564370.5403356437
L(12)L(\frac12) ≈\approx 0.54033564370.5403356437
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(−3.53+3.53i)T 1 + (-3.53 + 3.53i)T
good3 1+(6.39+6.39i)T+27iT2 1 + (6.39 + 6.39i)T + 27iT^{2}
7 1−30.7iT−343T2 1 - 30.7iT - 343T^{2}
11 1+(37.5−37.5i)T−1.33e3iT2 1 + (37.5 - 37.5i)T - 1.33e3iT^{2}
13 1+(22.1+22.1i)T+2.19e3iT2 1 + (22.1 + 22.1i)T + 2.19e3iT^{2}
17 1+27.4T+4.91e3T2 1 + 27.4T + 4.91e3T^{2}
19 1+(−46.7−46.7i)T+6.85e3iT2 1 + (-46.7 - 46.7i)T + 6.85e3iT^{2}
23 1+21.0iT−1.21e4T2 1 + 21.0iT - 1.21e4T^{2}
29 1+(61.1+61.1i)T+2.43e4iT2 1 + (61.1 + 61.1i)T + 2.43e4iT^{2}
31 1+75.2T+2.97e4T2 1 + 75.2T + 2.97e4T^{2}
37 1+(−236.+236.i)T−5.06e4iT2 1 + (-236. + 236. i)T - 5.06e4iT^{2}
41 1+317.iT−6.89e4T2 1 + 317. iT - 6.89e4T^{2}
43 1+(139.−139.i)T−7.95e4iT2 1 + (139. - 139. i)T - 7.95e4iT^{2}
47 1−428.T+1.03e5T2 1 - 428.T + 1.03e5T^{2}
53 1+(−140.+140.i)T−1.48e5iT2 1 + (-140. + 140. i)T - 1.48e5iT^{2}
59 1+(136.−136.i)T−2.05e5iT2 1 + (136. - 136. i)T - 2.05e5iT^{2}
61 1+(−65.1−65.1i)T+2.26e5iT2 1 + (-65.1 - 65.1i)T + 2.26e5iT^{2}
67 1+(692.+692.i)T+3.00e5iT2 1 + (692. + 692. i)T + 3.00e5iT^{2}
71 1+647.iT−3.57e5T2 1 + 647. iT - 3.57e5T^{2}
73 1−502.iT−3.89e5T2 1 - 502. iT - 3.89e5T^{2}
79 1+251.T+4.93e5T2 1 + 251.T + 4.93e5T^{2}
83 1+(482.+482.i)T+5.71e5iT2 1 + (482. + 482. i)T + 5.71e5iT^{2}
89 1−887.iT−7.04e5T2 1 - 887. iT - 7.04e5T^{2}
97 1−1.12e3T+9.12e5T2 1 - 1.12e3T + 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line