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Properties

Label 2-140-7.2-c1-0-1
Degree 22
Conductor 140140
Sign 0.701+0.712i0.701 + 0.712i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
Motivic weight 11
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)3-s + (0.5 − 0.866i)5-s + (2.5 + 0.866i)7-s + (1 − 1.73i)9-s + (−3 − 5.19i)11-s + 2·13-s − 0.999·15-s + (3 + 5.19i)17-s + (−4 + 6.92i)19-s + (−0.500 − 2.59i)21-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s − 5·27-s + 3·29-s + (−1 − 1.73i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.944 + 0.327i)7-s + (0.333 − 0.577i)9-s + (−0.904 − 1.56i)11-s + 0.554·13-s − 0.258·15-s + (0.727 + 1.26i)17-s + (−0.917 + 1.58i)19-s + (−0.109 − 0.566i)21-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s − 0.962·27-s + 0.557·29-s + (−0.179 − 0.311i)31-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((0.701+0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+1/2) L(s)=((0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.701+0.712i0.701 + 0.712i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ140(121,⋅)\chi_{140} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :1/2), 0.701+0.712i)(2,\ 140,\ (\ :1/2),\ 0.701 + 0.712i)

Particular Values

L(1)L(1) ≈\approx 1.02035−0.427612i1.02035 - 0.427612i
L(12)L(\frac12) ≈\approx 1.02035−0.427612i1.02035 - 0.427612i
L(32)L(\frac{3}{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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−2.5−0.866i)T 1 + (-2.5 - 0.866i)T
good3 1+(0.5+0.866i)T+(−1.5+2.59i)T2 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}
11 1+(3+5.19i)T+(−5.5+9.52i)T2 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1+(−3−5.19i)T+(−8.5+14.7i)T2 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2}
19 1+(4−6.92i)T+(−9.5−16.4i)T2 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.5−2.59i)T+(−11.5−19.9i)T2 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1+(1+1.73i)T+(−15.5+26.8i)T2 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1+(4−6.92i)T+(−18.5−32.0i)T2 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1−5T+43T2 1 - 5T + 43T^{2}
47 1+(−23.5−40.7i)T2 1 + (-23.5 - 40.7i)T^{2}
53 1+(6+10.3i)T+(−26.5+45.8i)T2 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(−0.5+0.866i)T+(−30.5−52.8i)T2 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.5−6.06i)T+(−33.5+58.0i)T2 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1+(−5−8.66i)T+(−36.5+63.2i)T2 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2+3.46i)T+(−39.5−68.4i)T2 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2}
83 1−3T+83T2 1 - 3T + 83T^{2}
89 1+(−1.5+2.59i)T+(−44.5−77.0i)T2 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2}
97 1+10T+97T2 1 + 10T + 97T^{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

−12.92537522565612462755954713376, −12.12519234019566390850642585439, −11.08619625972759325409613725196, −10.10354980205228988436042381203, −8.444227188983535485232190633415, −8.064615327217175668888713984766, −6.20864127638802650703356683793, −5.53064951770157521542506418499, −3.72731121181989691548103376156, −1.50858264136031691399360890224, 2.30107921941669512543073014083, 4.48615001777838913573719204557, 5.17040697582510466059222722227, 6.99365084362842027402329299925, 7.82713327274107256128074261923, 9.338167679084910960978160382074, 10.49487441071752086039410885698, 10.90730112731691485270055581781, 12.22731696367484953935176051160, 13.36706048412841757629063522976

Graph of the ZZ-function along the critical line