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Properties

Label 2-980-28.27-c1-0-53
Degree 22
Conductor 980980
Sign −0.785+0.619i-0.785 + 0.619i
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)=((−0.785+0.619i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.785+0.619i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.785+0.619i-0.785 + 0.619i
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), −0.785+0.619i)(2,\ 980,\ (\ :1/2),\ -0.785 + 0.619i)

Particular Values

L(1)L(1) ≈\approx 0.312036−0.899378i0.312036 - 0.899378i
L(12)L(\frac12) ≈\approx 0.312036−0.899378i0.312036 - 0.899378i
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

−9.398039342058804405926619636439, −9.001581757431571428928377983132, −8.284287962621193688575486744148, −7.49898464659585544098036878209, −6.46174703444166480601214177759, −5.09335215355765725525007185984, −4.03402205667223555923718277166, −2.97858776054495405351088606430, −2.13514602466028987641033619140, −0.50047095410346686000349122968, 1.62773461456053424638660875729, 2.87733769059205109424810061889, 4.10201619632146522511133608055, 5.48320488474189018982555115934, 6.11694846331530908804166289852, 7.16931345808113103883676096728, 7.987831287455493907600767967893, 8.425115569912232471175546283218, 9.494810216604026630086914747495, 10.00121686978251175011328366416

Graph of the ZZ-function along the critical line