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Properties

Label 2-980-28.27-c1-0-40
Degree 22
Conductor 980980
Sign 0.802−0.596i0.802 - 0.596i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (−1.40 − 0.153i)2-s + 3.02·3-s + (1.95 + 0.432i)4-s + i·5-s + (−4.25 − 0.465i)6-s + (−2.67 − 0.908i)8-s + 6.16·9-s + (0.153 − 1.40i)10-s + 1.19i·11-s + (5.91 + 1.30i)12-s + 4.83i·13-s + 3.02i·15-s + (3.62 + 1.68i)16-s − 2.54i·17-s + (−8.66 − 0.948i)18-s − 1.42·19-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s + 1.74·3-s + (0.976 + 0.216i)4-s + 0.447i·5-s + (−1.73 − 0.190i)6-s + (−0.946 − 0.321i)8-s + 2.05·9-s + (0.0486 − 0.444i)10-s + 0.360i·11-s + (1.70 + 0.378i)12-s + 1.34i·13-s + 0.781i·15-s + (0.906 + 0.422i)16-s − 0.617i·17-s + (−2.04 − 0.223i)18-s − 0.326·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.802−0.596i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.802−0.596i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.802−0.596i0.802 - 0.596i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(391,⋅)\chi_{980} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.802−0.596i)(2,\ 980,\ (\ :1/2),\ 0.802 - 0.596i)

Particular Values

L(1)L(1) ≈\approx 1.82499+0.603782i1.82499 + 0.603782i
L(12)L(\frac12) ≈\approx 1.82499+0.603782i1.82499 + 0.603782i
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+(1.40+0.153i)T 1 + (1.40 + 0.153i)T
5 1−iT 1 - iT
7 1 1
good3 1−3.02T+3T2 1 - 3.02T + 3T^{2}
11 1−1.19iT−11T2 1 - 1.19iT - 11T^{2}
13 1−4.83iT−13T2 1 - 4.83iT - 13T^{2}
17 1+2.54iT−17T2 1 + 2.54iT - 17T^{2}
19 1+1.42T+19T2 1 + 1.42T + 19T^{2}
23 1−5.80iT−23T2 1 - 5.80iT - 23T^{2}
29 1−0.774T+29T2 1 - 0.774T + 29T^{2}
31 1−6.63T+31T2 1 - 6.63T + 31T^{2}
37 1−5.10T+37T2 1 - 5.10T + 37T^{2}
41 1+7.46iT−41T2 1 + 7.46iT - 41T^{2}
43 1−1.38iT−43T2 1 - 1.38iT - 43T^{2}
47 1−1.07T+47T2 1 - 1.07T + 47T^{2}
53 1−3.36T+53T2 1 - 3.36T + 53T^{2}
59 1+9.88T+59T2 1 + 9.88T + 59T^{2}
61 1−9.59iT−61T2 1 - 9.59iT - 61T^{2}
67 1+10.5iT−67T2 1 + 10.5iT - 67T^{2}
71 1+16.3iT−71T2 1 + 16.3iT - 71T^{2}
73 1+0.107iT−73T2 1 + 0.107iT - 73T^{2}
79 1+10.7iT−79T2 1 + 10.7iT - 79T^{2}
83 1+15.8T+83T2 1 + 15.8T + 83T^{2}
89 1+3.94iT−89T2 1 + 3.94iT - 89T^{2}
97 1−8.71iT−97T2 1 - 8.71iT - 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

−9.699452760896838609656581637379, −9.260260649004423046289981085675, −8.589574969358358576251086177275, −7.63099769066164382859594522617, −7.21440965908745223949938195387, −6.26845818759827740025699580613, −4.41945013240139207296101046984, −3.38321593957579651047895187745, −2.49249105085878610547722807393, −1.65139828917744795609734882896, 1.08073342314369534068772876422, 2.43879404050506736045606454806, 3.12883115208931453676482963645, 4.37113252473887083197331699205, 5.84404508221303768162116907714, 6.88050943925133463975912911089, 7.985287711924022059502659491518, 8.259584976736282643070291294499, 8.823089255960218651505348901794, 9.830331712348421751662704564157

Graph of the ZZ-function along the critical line