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Properties

Label 2-280-7.4-c3-0-7
Degree 22
Conductor 280280
Sign −0.991+0.126i-0.991 + 0.126i
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  + (−3.5 + 6.06i)3-s + (2.5 + 4.33i)5-s + (3.5 + 18.1i)7-s + (−11 − 19.0i)9-s + (−29 + 50.2i)11-s + 82·13-s − 35·15-s + (−25 + 43.3i)17-s + (−32 − 55.4i)19-s + (−122.5 − 42.4i)21-s + (55.5 + 96.1i)23-s + (−12.5 + 21.6i)25-s − 35.0·27-s + 103·29-s + (65 − 112. i)31-s + ⋯
L(s)  = 1  + (−0.673 + 1.16i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.407 − 0.705i)9-s + (−0.794 + 1.37i)11-s + 1.74·13-s − 0.602·15-s + (−0.356 + 0.617i)17-s + (−0.386 − 0.669i)19-s + (−1.27 − 0.440i)21-s + (0.503 + 0.871i)23-s + (−0.100 + 0.173i)25-s − 0.249·27-s + 0.659·29-s + (0.376 − 0.652i)31-s + ⋯

Functional equation

Λ(s)=(280s/2 ΓC(s) L(s)=((−0.991+0.126i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(280s/2 ΓC(s+3/2) L(s)=((−0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 280280    =    23⋅5⋅72^{3} \cdot 5 \cdot 7
Sign: −0.991+0.126i-0.991 + 0.126i
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), −0.991+0.126i)(2,\ 280,\ (\ :3/2),\ -0.991 + 0.126i)

Particular Values

L(2)L(2) ≈\approx 1.1824615651.182461565
L(12)L(\frac12) ≈\approx 1.1824615651.182461565
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+(−3.5−18.1i)T 1 + (-3.5 - 18.1i)T
good3 1+(3.5−6.06i)T+(−13.5−23.3i)T2 1 + (3.5 - 6.06i)T + (-13.5 - 23.3i)T^{2}
11 1+(29−50.2i)T+(−665.5−1.15e3i)T2 1 + (29 - 50.2i)T + (-665.5 - 1.15e3i)T^{2}
13 1−82T+2.19e3T2 1 - 82T + 2.19e3T^{2}
17 1+(25−43.3i)T+(−2.45e3−4.25e3i)T2 1 + (25 - 43.3i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(32+55.4i)T+(−3.42e3+5.94e3i)T2 1 + (32 + 55.4i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−55.5−96.1i)T+(−6.08e3+1.05e4i)T2 1 + (-55.5 - 96.1i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1−103T+2.43e4T2 1 - 103T + 2.43e4T^{2}
31 1+(−65+112.i)T+(−1.48e4−2.57e4i)T2 1 + (-65 + 112. i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(188+325.i)T+(−2.53e4+4.38e4i)T2 1 + (188 + 325. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+307T+6.89e4T2 1 + 307T + 6.89e4T^{2}
43 1+197T+7.95e4T2 1 + 197T + 7.95e4T^{2}
47 1+(60+103.i)T+(−5.19e4+8.99e4i)T2 1 + (60 + 103. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−254+439.i)T+(−7.44e4−1.28e5i)T2 1 + (-254 + 439. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(300−519.i)T+(−1.02e5−1.77e5i)T2 1 + (300 - 519. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(−82.5−142.i)T+(−1.13e5+1.96e5i)T2 1 + (-82.5 - 142. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−316.5+548.i)T+(−1.50e5−2.60e5i)T2 1 + (-316.5 + 548. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1−840T+3.57e5T2 1 - 840T + 3.57e5T^{2}
73 1+(303−524.i)T+(−1.94e5−3.36e5i)T2 1 + (303 - 524. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−658−1.13e3i)T+(−2.46e5+4.26e5i)T2 1 + (-658 - 1.13e3i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−61T+5.71e5T2 1 - 61T + 5.71e5T^{2}
89 1+(−93.5−161.i)T+(−3.52e5+6.10e5i)T2 1 + (-93.5 - 161. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−406T+9.12e5T2 1 - 406T + 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

−11.59560752761809697768659222039, −10.89034930035001721152852817513, −10.18100299253983700917528019161, −9.248393436704561328086756911518, −8.284062543157981878091260833824, −6.76478147636842431553882527762, −5.67745049379818177983647645963, −4.89795716314073745381313288440, −3.69857540536067335720712030590, −2.08338995306755589223515431297, 0.52185977981391718091680818150, 1.41318428586734288907352820178, 3.36276871556921022259308770403, 4.95362632355586402340829423380, 6.15926143775471184016708559026, 6.73044293522881481754444403738, 8.080120498539775468298745083400, 8.612740752605213903175287420079, 10.36271321260575420694687390688, 10.99964615122909534875762087810

Graph of the ZZ-function along the critical line