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Properties

Label 2-560-7.4-c3-0-16
Degree 22
Conductor 560560
Sign −0.266−0.963i-0.266 - 0.963i
Analytic cond. 33.041033.0410
Root an. cond. 5.748135.74813
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  + (−1 + 1.73i)3-s + (−2.5 − 4.33i)5-s + (14 + 12.1i)7-s + (11.5 + 19.9i)9-s + (−22.5 + 38.9i)11-s + 59·13-s + 10·15-s + (27 − 46.7i)17-s + (−60.5 − 104. i)19-s + (−35 + 12.1i)21-s + (34.5 + 59.7i)23-s + (−12.5 + 21.6i)25-s − 100·27-s − 162·29-s + (−44 + 76.2i)31-s + ⋯
L(s)  = 1  + (−0.192 + 0.333i)3-s + (−0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (0.425 + 0.737i)9-s + (−0.616 + 1.06i)11-s + 1.25·13-s + 0.172·15-s + (0.385 − 0.667i)17-s + (−0.730 − 1.26i)19-s + (−0.363 + 0.125i)21-s + (0.312 + 0.541i)23-s + (−0.100 + 0.173i)25-s − 0.712·27-s − 1.03·29-s + (−0.254 + 0.441i)31-s + ⋯

Functional equation

Λ(s)=(560s/2 ΓC(s) L(s)=((−0.266−0.963i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(560s/2 ΓC(s+3/2) L(s)=((−0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24⋅5⋅72^{4} \cdot 5 \cdot 7
Sign: −0.266−0.963i-0.266 - 0.963i
Analytic conductor: 33.041033.0410
Root analytic conductor: 5.748135.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ560(81,⋅)\chi_{560} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 560, ( :3/2), −0.266−0.963i)(2,\ 560,\ (\ :3/2),\ -0.266 - 0.963i)

Particular Values

L(2)L(2) ≈\approx 1.6661666351.666166635
L(12)L(\frac12) ≈\approx 1.6661666351.666166635
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+(2.5+4.33i)T 1 + (2.5 + 4.33i)T
7 1+(−14−12.1i)T 1 + (-14 - 12.1i)T
good3 1+(1−1.73i)T+(−13.5−23.3i)T2 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2}
11 1+(22.5−38.9i)T+(−665.5−1.15e3i)T2 1 + (22.5 - 38.9i)T + (-665.5 - 1.15e3i)T^{2}
13 1−59T+2.19e3T2 1 - 59T + 2.19e3T^{2}
17 1+(−27+46.7i)T+(−2.45e3−4.25e3i)T2 1 + (-27 + 46.7i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(60.5+104.i)T+(−3.42e3+5.94e3i)T2 1 + (60.5 + 104. i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−34.5−59.7i)T+(−6.08e3+1.05e4i)T2 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+162T+2.43e4T2 1 + 162T + 2.43e4T^{2}
31 1+(44−76.2i)T+(−1.48e4−2.57e4i)T2 1 + (44 - 76.2i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−129.5−224.i)T+(−2.53e4+4.38e4i)T2 1 + (-129.5 - 224. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−195T+6.89e4T2 1 - 195T + 6.89e4T^{2}
43 1−286T+7.95e4T2 1 - 286T + 7.95e4T^{2}
47 1+(−22.5−38.9i)T+(−5.19e4+8.99e4i)T2 1 + (-22.5 - 38.9i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(298.5−517.i)T+(−7.44e4−1.28e5i)T2 1 + (298.5 - 517. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(180−311.i)T+(−1.02e5−1.77e5i)T2 1 + (180 - 311. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(196+339.i)T+(−1.13e5+1.96e5i)T2 1 + (196 + 339. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(140−242.i)T+(−1.50e5−2.60e5i)T2 1 + (140 - 242. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+48T+3.57e5T2 1 + 48T + 3.57e5T^{2}
73 1+(334−578.i)T+(−1.94e5−3.36e5i)T2 1 + (334 - 578. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−391−677.i)T+(−2.46e5+4.26e5i)T2 1 + (-391 - 677. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+768T+5.71e5T2 1 + 768T + 5.71e5T^{2}
89 1+(−597−1.03e3i)T+(−3.52e5+6.10e5i)T2 1 + (-597 - 1.03e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−902T+9.12e5T2 1 - 902T + 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

−10.91796874532800871770764223549, −9.654501098455661572097993301956, −8.910069059219240319585256136259, −7.927833583667584658875731710814, −7.22540348026008709696611574359, −5.77707550062993724905007182985, −4.92662207633951749705784223142, −4.27556526153179904551614671538, −2.60924418659676248201970716497, −1.39978659894588040544829320853, 0.54509718477450021843362418057, 1.71456253059332330903925781118, 3.48954687785082697490935015396, 4.13175858058002997581577793331, 5.76647327660066276511160302097, 6.30667999154460271280950449961, 7.56875381283536026398297967529, 8.118153379539610668149114124838, 9.114871375564382126972698385374, 10.48363156053982916687705063038

Graph of the ZZ-function along the critical line