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Properties

Label 2-980-28.27-c1-0-10
Degree 22
Conductor 980980
Sign −0.949−0.314i-0.949 - 0.314i
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  + (1.17 + 0.784i)2-s − 2.99·3-s + (0.767 + 1.84i)4-s − i·5-s + (−3.52 − 2.35i)6-s + (−0.546 + 2.77i)8-s + 5.98·9-s + (0.784 − 1.17i)10-s − 2.23i·11-s + (−2.30 − 5.53i)12-s + 3.17i·13-s + 2.99i·15-s + (−2.82 + 2.83i)16-s + 3.44i·17-s + (7.04 + 4.70i)18-s + 2.05·19-s + ⋯
L(s)  = 1  + (0.831 + 0.555i)2-s − 1.73·3-s + (0.383 + 0.923i)4-s − 0.447i·5-s + (−1.43 − 0.960i)6-s + (−0.193 + 0.981i)8-s + 1.99·9-s + (0.248 − 0.372i)10-s − 0.674i·11-s + (−0.664 − 1.59i)12-s + 0.879i·13-s + 0.774i·15-s + (−0.705 + 0.708i)16-s + 0.835i·17-s + (1.66 + 1.10i)18-s + 0.470·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.949−0.314i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.949−0.314i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.949−0.314i-0.949 - 0.314i
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), −0.949−0.314i)(2,\ 980,\ (\ :1/2),\ -0.949 - 0.314i)

Particular Values

L(1)L(1) ≈\approx 0.132916+0.824361i0.132916 + 0.824361i
L(12)L(\frac12) ≈\approx 0.132916+0.824361i0.132916 + 0.824361i
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+(−1.17−0.784i)T 1 + (-1.17 - 0.784i)T
5 1+iT 1 + iT
7 1 1
good3 1+2.99T+3T2 1 + 2.99T + 3T^{2}
11 1+2.23iT−11T2 1 + 2.23iT - 11T^{2}
13 1−3.17iT−13T2 1 - 3.17iT - 13T^{2}
17 1−3.44iT−17T2 1 - 3.44iT - 17T^{2}
19 1−2.05T+19T2 1 - 2.05T + 19T^{2}
23 1−2.66iT−23T2 1 - 2.66iT - 23T^{2}
29 1+7.38T+29T2 1 + 7.38T + 29T^{2}
31 1+4.89T+31T2 1 + 4.89T + 31T^{2}
37 1+11.1T+37T2 1 + 11.1T + 37T^{2}
41 1−1.46iT−41T2 1 - 1.46iT - 41T^{2}
43 1−9.95iT−43T2 1 - 9.95iT - 43T^{2}
47 1−6.12T+47T2 1 - 6.12T + 47T^{2}
53 1−4.65T+53T2 1 - 4.65T + 53T^{2}
59 1+7.11T+59T2 1 + 7.11T + 59T^{2}
61 1+2.53iT−61T2 1 + 2.53iT - 61T^{2}
67 1+0.0527iT−67T2 1 + 0.0527iT - 67T^{2}
71 1−0.212iT−71T2 1 - 0.212iT - 71T^{2}
73 1−14.8iT−73T2 1 - 14.8iT - 73T^{2}
79 1+0.461iT−79T2 1 + 0.461iT - 79T^{2}
83 1+10.9T+83T2 1 + 10.9T + 83T^{2}
89 1−7.02iT−89T2 1 - 7.02iT - 89T^{2}
97 1+0.185iT−97T2 1 + 0.185iT - 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

−10.85230036360661118734562108348, −9.594332140048011157833369302602, −8.580816160976282103983759841109, −7.44492678665017593968998602868, −6.74008121142595906116120336398, −5.80301194002328497904525810061, −5.45647833632790441925321758347, −4.46300159390410103788749440403, −3.62267319303442031053311196107, −1.61950641206102807297606897023, 0.36191551942283257256459630992, 1.90109509951191267894198854107, 3.38554446507200618368307105033, 4.49143690051416194271405350471, 5.40448221682091403662557304794, 5.75946118034173763067453244437, 6.98418379003159912063262678272, 7.28682967920298782733457027361, 9.214157383662037860093830639089, 10.20276415212209195287531093926

Graph of the ZZ-function along the critical line