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Properties

Label 2-140-7.3-c4-0-0
Degree 22
Conductor 140140
Sign 0.280−0.959i0.280 - 0.959i
Analytic cond. 14.471714.4717
Root an. cond. 3.804183.80418
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.02 − 4.63i)3-s + (−9.68 + 5.59i)5-s + (−5.50 − 48.6i)7-s + (2.44 + 4.22i)9-s + (−62.7 + 108. i)11-s − 130. i·13-s + 103.·15-s + (251. + 145. i)17-s + (−364. + 210. i)19-s + (−181. + 416. i)21-s + (417. + 723. i)23-s + (62.5 − 108. i)25-s + 705. i·27-s + 689.·29-s + (155. + 89.5i)31-s + ⋯
L(s)  = 1  + (−0.891 − 0.514i)3-s + (−0.387 + 0.223i)5-s + (−0.112 − 0.993i)7-s + (0.0301 + 0.0522i)9-s + (−0.518 + 0.898i)11-s − 0.770i·13-s + 0.460·15-s + (0.871 + 0.503i)17-s + (−1.01 + 0.583i)19-s + (−0.411 + 0.943i)21-s + (0.789 + 1.36i)23-s + (0.100 − 0.173i)25-s + 0.967i·27-s + 0.820·29-s + (0.161 + 0.0931i)31-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((0.280−0.959i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+2) L(s)=((0.280−0.959i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.280−0.959i0.280 - 0.959i
Analytic conductor: 14.471714.4717
Root analytic conductor: 3.804183.80418
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ140(101,⋅)\chi_{140} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :2), 0.280−0.959i)(2,\ 140,\ (\ :2),\ 0.280 - 0.959i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.468785+0.351429i0.468785 + 0.351429i
L(12)L(\frac12) ≈\approx 0.468785+0.351429i0.468785 + 0.351429i
L(3)L(3) 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+(9.68−5.59i)T 1 + (9.68 - 5.59i)T
7 1+(5.50+48.6i)T 1 + (5.50 + 48.6i)T
good3 1+(8.02+4.63i)T+(40.5+70.1i)T2 1 + (8.02 + 4.63i)T + (40.5 + 70.1i)T^{2}
11 1+(62.7−108.i)T+(−7.32e3−1.26e4i)T2 1 + (62.7 - 108. i)T + (-7.32e3 - 1.26e4i)T^{2}
13 1+130.iT−2.85e4T2 1 + 130. iT - 2.85e4T^{2}
17 1+(−251.−145.i)T+(4.17e4+7.23e4i)T2 1 + (-251. - 145. i)T + (4.17e4 + 7.23e4i)T^{2}
19 1+(364.−210.i)T+(6.51e4−1.12e5i)T2 1 + (364. - 210. i)T + (6.51e4 - 1.12e5i)T^{2}
23 1+(−417.−723.i)T+(−1.39e5+2.42e5i)T2 1 + (-417. - 723. i)T + (-1.39e5 + 2.42e5i)T^{2}
29 1−689.T+7.07e5T2 1 - 689.T + 7.07e5T^{2}
31 1+(−155.−89.5i)T+(4.61e5+7.99e5i)T2 1 + (-155. - 89.5i)T + (4.61e5 + 7.99e5i)T^{2}
37 1+(−1.18e3−2.05e3i)T+(−9.37e5+1.62e6i)T2 1 + (-1.18e3 - 2.05e3i)T + (-9.37e5 + 1.62e6i)T^{2}
41 1+8.95iT−2.82e6T2 1 + 8.95iT - 2.82e6T^{2}
43 1+3.41e3T+3.41e6T2 1 + 3.41e3T + 3.41e6T^{2}
47 1+(−1.79e3+1.03e3i)T+(2.43e6−4.22e6i)T2 1 + (-1.79e3 + 1.03e3i)T + (2.43e6 - 4.22e6i)T^{2}
53 1+(1.18e3−2.06e3i)T+(−3.94e6−6.83e6i)T2 1 + (1.18e3 - 2.06e3i)T + (-3.94e6 - 6.83e6i)T^{2}
59 1+(1.98e3+1.14e3i)T+(6.05e6+1.04e7i)T2 1 + (1.98e3 + 1.14e3i)T + (6.05e6 + 1.04e7i)T^{2}
61 1+(−2.37e3+1.36e3i)T+(6.92e6−1.19e7i)T2 1 + (-2.37e3 + 1.36e3i)T + (6.92e6 - 1.19e7i)T^{2}
67 1+(−945.+1.63e3i)T+(−1.00e7−1.74e7i)T2 1 + (-945. + 1.63e3i)T + (-1.00e7 - 1.74e7i)T^{2}
71 1+8.10e3T+2.54e7T2 1 + 8.10e3T + 2.54e7T^{2}
73 1+(−831.−480.i)T+(1.41e7+2.45e7i)T2 1 + (-831. - 480. i)T + (1.41e7 + 2.45e7i)T^{2}
79 1+(−4.18e3−7.25e3i)T+(−1.94e7+3.37e7i)T2 1 + (-4.18e3 - 7.25e3i)T + (-1.94e7 + 3.37e7i)T^{2}
83 1+7.33e3iT−4.74e7T2 1 + 7.33e3iT - 4.74e7T^{2}
89 1+(−1.69e3+975.i)T+(3.13e7−5.43e7i)T2 1 + (-1.69e3 + 975. i)T + (3.13e7 - 5.43e7i)T^{2}
97 1−4.68e3iT−8.85e7T2 1 - 4.68e3iT - 8.85e7T^{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

−12.60648788886325522053966704844, −11.72352122340749936315840543594, −10.64842134840740146468836070585, −9.950987610647614498794314695907, −8.108800529684535481950148117881, −7.23410312384142115351980653643, −6.22603207804117927606194135083, −4.91169901375682631391714915375, −3.39460788872667980018130615346, −1.18205696290835281035465503994, 0.30377053107423277089112898293, 2.70506451057036517852497020315, 4.52060498522021747755219085762, 5.47514287965341984237746562993, 6.52675282910896993787075299316, 8.202299542390947710662456371216, 9.078554729695972401171510174245, 10.42989849451324258951379750356, 11.28436939994337143148021203735, 12.03280336967888828585021940736

Graph of the ZZ-function along the critical line