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Properties

Label 2-140-140.39-c0-0-1
Degree 22
Conductor 140140
Sign −0.0633+0.997i-0.0633 + 0.997i
Analytic cond. 0.06986910.0698691
Root an. cond. 0.2643270.264327
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + 0.999·20-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)10-s + (0.499 + 0.866i)12-s + 0.999·14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + 0.999·20-s + (0.499 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.499 − 0.866i)24-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((−0.0633+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s) L(s)=((−0.0633+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: −0.0633+0.997i-0.0633 + 0.997i
Analytic conductor: 0.06986910.0698691
Root analytic conductor: 0.2643270.264327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ140(39,⋅)\chi_{140} (39, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :0), −0.0633+0.997i)(2,\ 140,\ (\ :0),\ -0.0633 + 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.51422616170.5142261617
L(12)L(\frac12) ≈\approx 0.51422616170.5142261617
L(1)L(1) 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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good3 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
23 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1+T+T2 1 + T + T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1−T2 1 - T^{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

−13.02974905280989370564192259687, −12.17860024544497028843624594850, −11.42040436361054528192267196081, −9.877769206049269076046763875976, −8.868601555853835613352467318640, −8.180747780401275503283824113235, −7.10387591591902298203109404153, −5.13376199973804018896584366820, −3.41763793665203501307430262132, −1.84798733176887999444039248112, 3.43782950479146480647989201129, 4.57530209214505313204504732398, 6.41661967871024913557960236053, 7.26104126294525014046992686963, 8.416143973274795439126093561837, 9.614591293890291152644650722770, 10.27748392754102519390547451999, 11.16716525752643526132281201912, 12.96999685799762280518674606496, 14.17587321847294524728955915446

Graph of the ZZ-function along the critical line