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Properties

Label 2-980-28.27-c1-0-25
Degree 22
Conductor 980980
Sign 0.725−0.688i0.725 - 0.688i
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  + (−0.976 + 1.02i)2-s − 1.11·3-s + (−0.0916 − 1.99i)4-s − i·5-s + (1.08 − 1.13i)6-s + (2.13 + 1.85i)8-s − 1.76·9-s + (1.02 + 0.976i)10-s + 1.71i·11-s + (0.101 + 2.22i)12-s + 2.45i·13-s + 1.11i·15-s + (−3.98 + 0.366i)16-s − 6.21i·17-s + (1.72 − 1.80i)18-s + 0.216·19-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s − 0.642·3-s + (−0.0458 − 0.998i)4-s − 0.447i·5-s + (0.443 − 0.464i)6-s + (0.753 + 0.656i)8-s − 0.587·9-s + (0.323 + 0.308i)10-s + 0.516i·11-s + (0.0294 + 0.641i)12-s + 0.682i·13-s + 0.287i·15-s + (−0.995 + 0.0915i)16-s − 1.50i·17-s + (0.405 − 0.424i)18-s + 0.0496·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.725−0.688i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.725−0.688i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.725−0.688i0.725 - 0.688i
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), 0.725−0.688i)(2,\ 980,\ (\ :1/2),\ 0.725 - 0.688i)

Particular Values

L(1)L(1) ≈\approx 0.676083+0.269857i0.676083 + 0.269857i
L(12)L(\frac12) ≈\approx 0.676083+0.269857i0.676083 + 0.269857i
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+(0.976−1.02i)T 1 + (0.976 - 1.02i)T
5 1+iT 1 + iT
7 1 1
good3 1+1.11T+3T2 1 + 1.11T + 3T^{2}
11 1−1.71iT−11T2 1 - 1.71iT - 11T^{2}
13 1−2.45iT−13T2 1 - 2.45iT - 13T^{2}
17 1+6.21iT−17T2 1 + 6.21iT - 17T^{2}
19 1−0.216T+19T2 1 - 0.216T + 19T^{2}
23 1−6.56iT−23T2 1 - 6.56iT - 23T^{2}
29 1+2.47T+29T2 1 + 2.47T + 29T^{2}
31 1+0.163T+31T2 1 + 0.163T + 31T^{2}
37 1−7.69T+37T2 1 - 7.69T + 37T^{2}
41 1+8.34iT−41T2 1 + 8.34iT - 41T^{2}
43 1−1.89iT−43T2 1 - 1.89iT - 43T^{2}
47 1−11.7T+47T2 1 - 11.7T + 47T^{2}
53 1−13.0T+53T2 1 - 13.0T + 53T^{2}
59 1−4.28T+59T2 1 - 4.28T + 59T^{2}
61 1+7.00iT−61T2 1 + 7.00iT - 61T^{2}
67 1−5.17iT−67T2 1 - 5.17iT - 67T^{2}
71 1−5.04iT−71T2 1 - 5.04iT - 71T^{2}
73 1−7.61iT−73T2 1 - 7.61iT - 73T^{2}
79 1−15.9iT−79T2 1 - 15.9iT - 79T^{2}
83 1−5.47T+83T2 1 - 5.47T + 83T^{2}
89 1−1.78iT−89T2 1 - 1.78iT - 89T^{2}
97 1−10.5iT−97T2 1 - 10.5iT - 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

−9.782300964512576161978367360000, −9.301442693751404294254735793320, −8.502311282169780382131056141130, −7.42298805762126748563729910630, −6.89876503348542375209896756499, −5.69119590849188556045558914300, −5.28479720196408038899267283925, −4.19995297559564467063223682289, −2.34656423077108842520966640066, −0.814977143771299697532100388009, 0.70135645215088026144002283737, 2.35094467339100054370109045033, 3.32801924900803614522287636659, 4.40155148710351467142479416817, 5.77392887461719931482281380318, 6.42621184441108168296708153430, 7.61035042892991348648639205167, 8.378695127186582378269080426837, 9.037954346340147061080255920562, 10.38539594375642733223045739467

Graph of the ZZ-function along the critical line