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Properties

Label 2-280-7.4-c3-0-8
Degree 22
Conductor 280280
Sign 0.335−0.942i0.335 - 0.942i
Analytic cond. 16.520516.5205
Root an. cond. 4.064544.06454
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  + (1.23 − 2.14i)3-s + (2.5 + 4.33i)5-s + (17.8 + 5.09i)7-s + (10.4 + 18.0i)9-s + (−26.0 + 45.0i)11-s − 53.4·13-s + 12.3·15-s + (−2.44 + 4.23i)17-s + (−51.1 − 88.5i)19-s + (32.9 − 31.8i)21-s + (105. + 183. i)23-s + (−12.5 + 21.6i)25-s + 118.·27-s + 0.173·29-s + (−123. + 214. i)31-s + ⋯
L(s)  = 1  + (0.238 − 0.412i)3-s + (0.223 + 0.387i)5-s + (0.961 + 0.275i)7-s + (0.386 + 0.669i)9-s + (−0.712 + 1.23i)11-s − 1.13·13-s + 0.213·15-s + (−0.0349 + 0.0604i)17-s + (−0.617 − 1.06i)19-s + (0.342 − 0.331i)21-s + (0.960 + 1.66i)23-s + (−0.100 + 0.173i)25-s + 0.844·27-s + 0.00110·29-s + (−0.716 + 1.24i)31-s + ⋯

Functional equation

Λ(s)=(280s/2 ΓC(s) L(s)=((0.335−0.942i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(280s/2 ΓC(s+3/2) L(s)=((0.335−0.942i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 280280    =    23⋅5⋅72^{3} \cdot 5 \cdot 7
Sign: 0.335−0.942i0.335 - 0.942i
Analytic conductor: 16.520516.5205
Root analytic conductor: 4.064544.06454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ280(81,⋅)\chi_{280} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 280, ( :3/2), 0.335−0.942i)(2,\ 280,\ (\ :3/2),\ 0.335 - 0.942i)

Particular Values

L(2)L(2) ≈\approx 1.9103212781.910321278
L(12)L(\frac12) ≈\approx 1.9103212781.910321278
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+(−17.8−5.09i)T 1 + (-17.8 - 5.09i)T
good3 1+(−1.23+2.14i)T+(−13.5−23.3i)T2 1 + (-1.23 + 2.14i)T + (-13.5 - 23.3i)T^{2}
11 1+(26.0−45.0i)T+(−665.5−1.15e3i)T2 1 + (26.0 - 45.0i)T + (-665.5 - 1.15e3i)T^{2}
13 1+53.4T+2.19e3T2 1 + 53.4T + 2.19e3T^{2}
17 1+(2.44−4.23i)T+(−2.45e3−4.25e3i)T2 1 + (2.44 - 4.23i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(51.1+88.5i)T+(−3.42e3+5.94e3i)T2 1 + (51.1 + 88.5i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−105.−183.i)T+(−6.08e3+1.05e4i)T2 1 + (-105. - 183. i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1−0.173T+2.43e4T2 1 - 0.173T + 2.43e4T^{2}
31 1+(123.−214.i)T+(−1.48e4−2.57e4i)T2 1 + (123. - 214. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−41.5−71.9i)T+(−2.53e4+4.38e4i)T2 1 + (-41.5 - 71.9i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1−210.T+6.89e4T2 1 - 210.T + 6.89e4T^{2}
43 1−89.9T+7.95e4T2 1 - 89.9T + 7.95e4T^{2}
47 1+(18.8+32.6i)T+(−5.19e4+8.99e4i)T2 1 + (18.8 + 32.6i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−356.+618.i)T+(−7.44e4−1.28e5i)T2 1 + (-356. + 618. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(119.−206.i)T+(−1.02e5−1.77e5i)T2 1 + (119. - 206. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−130.−226.i)T+(−1.13e5+1.96e5i)T2 1 + (-130. - 226. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(423.−733.i)T+(−1.50e5−2.60e5i)T2 1 + (423. - 733. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−686.T+3.57e5T2 1 - 686.T + 3.57e5T^{2}
73 1+(−155.+269.i)T+(−1.94e5−3.36e5i)T2 1 + (-155. + 269. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(254.+440.i)T+(−2.46e5+4.26e5i)T2 1 + (254. + 440. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+633.T+5.71e5T2 1 + 633.T + 5.71e5T^{2}
89 1+(479.+829.i)T+(−3.52e5+6.10e5i)T2 1 + (479. + 829. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−962.T+9.12e5T2 1 - 962.T + 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

−11.57121845565739190350867515996, −10.66680205100925353795517249525, −9.790086460421892904833060610456, −8.655520748352436388146515985180, −7.31406671718290455432732494897, −7.26600110996367685285918340619, −5.30232263782197121507469567077, −4.65094116421294233764464987949, −2.62529506315183127638918697631, −1.75782551795271121432517834976, 0.70723576501709867875047204968, 2.46774947836751756196771241530, 4.00796900132496428401062543852, 4.97120053413469239288410663248, 6.11088273551592193558113754276, 7.51962968562301655086111691644, 8.416891860037571849467406987237, 9.281239458006898871817407887919, 10.36839545615632263982904661258, 11.04022889591232232689563012031

Graph of the ZZ-function along the critical line