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Properties

Label 2-980-28.19-c1-0-24
Degree 22
Conductor 980980
Sign −0.997−0.0661i-0.997 - 0.0661i
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  + (−1.26 + 0.626i)2-s + (1.49 + 2.59i)3-s + (1.21 − 1.58i)4-s + (0.866 + 0.5i)5-s + (−3.52 − 2.35i)6-s + (−0.546 + 2.77i)8-s + (−2.99 + 5.18i)9-s + (−1.41 − 0.0915i)10-s + (−1.93 + 1.11i)11-s + (5.94 + 0.774i)12-s + 3.17i·13-s + 2.99i·15-s + (−1.04 − 3.86i)16-s + (2.98 − 1.72i)17-s + (0.548 − 8.45i)18-s + (−1.02 + 1.77i)19-s + ⋯
L(s)  = 1  + (−0.896 + 0.442i)2-s + (0.865 + 1.49i)3-s + (0.607 − 0.794i)4-s + (0.387 + 0.223i)5-s + (−1.43 − 0.960i)6-s + (−0.193 + 0.981i)8-s + (−0.998 + 1.72i)9-s + (−0.446 − 0.0289i)10-s + (−0.584 + 0.337i)11-s + (1.71 + 0.223i)12-s + 0.879i·13-s + 0.774i·15-s + (−0.261 − 0.965i)16-s + (0.723 − 0.417i)17-s + (0.129 − 1.99i)18-s + (−0.235 + 0.407i)19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.997−0.0661i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0661i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.997−0.0661i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.997−0.0661i-0.997 - 0.0661i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(411,⋅)\chi_{980} (411, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.997−0.0661i)(2,\ 980,\ (\ :1/2),\ -0.997 - 0.0661i)

Particular Values

L(1)L(1) ≈\approx 0.0415307+1.25389i0.0415307 + 1.25389i
L(12)L(\frac12) ≈\approx 0.0415307+1.25389i0.0415307 + 1.25389i
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+(1.26−0.626i)T 1 + (1.26 - 0.626i)T
5 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1 1
good3 1+(−1.49−2.59i)T+(−1.5+2.59i)T2 1 + (-1.49 - 2.59i)T + (-1.5 + 2.59i)T^{2}
11 1+(1.93−1.11i)T+(5.5−9.52i)T2 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2}
13 1−3.17iT−13T2 1 - 3.17iT - 13T^{2}
17 1+(−2.98+1.72i)T+(8.5−14.7i)T2 1 + (-2.98 + 1.72i)T + (8.5 - 14.7i)T^{2}
19 1+(1.02−1.77i)T+(−9.5−16.4i)T2 1 + (1.02 - 1.77i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.30+1.33i)T+(11.5+19.9i)T2 1 + (2.30 + 1.33i)T + (11.5 + 19.9i)T^{2}
29 1+7.38T+29T2 1 + 7.38T + 29T^{2}
31 1+(−2.44−4.23i)T+(−15.5+26.8i)T2 1 + (-2.44 - 4.23i)T + (-15.5 + 26.8i)T^{2}
37 1+(−5.59+9.69i)T+(−18.5−32.0i)T2 1 + (-5.59 + 9.69i)T + (-18.5 - 32.0i)T^{2}
41 1−1.46iT−41T2 1 - 1.46iT - 41T^{2}
43 1−9.95iT−43T2 1 - 9.95iT - 43T^{2}
47 1+(3.06−5.30i)T+(−23.5−40.7i)T2 1 + (3.06 - 5.30i)T + (-23.5 - 40.7i)T^{2}
53 1+(2.32+4.03i)T+(−26.5+45.8i)T2 1 + (2.32 + 4.03i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.55−6.16i)T+(−29.5+51.0i)T2 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.19−1.26i)T+(30.5+52.8i)T2 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2}
67 1+(0.0456−0.0263i)T+(33.5−58.0i)T2 1 + (0.0456 - 0.0263i)T + (33.5 - 58.0i)T^{2}
71 1−0.212iT−71T2 1 - 0.212iT - 71T^{2}
73 1+(−12.8+7.43i)T+(36.5−63.2i)T2 1 + (-12.8 + 7.43i)T + (36.5 - 63.2i)T^{2}
79 1+(−0.399−0.230i)T+(39.5+68.4i)T2 1 + (-0.399 - 0.230i)T + (39.5 + 68.4i)T^{2}
83 1+10.9T+83T2 1 + 10.9T + 83T^{2}
89 1+(6.07+3.51i)T+(44.5+77.0i)T2 1 + (6.07 + 3.51i)T + (44.5 + 77.0i)T^{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

−9.987296623294979579150235220896, −9.631631428670800153134728785154, −8.961798589884689184922409313669, −8.088693858042615801969920917762, −7.37002642283235167152235405244, −6.11882197541353967730184750742, −5.20495543921025010688870042465, −4.26267514361640666957703027488, −3.00554740320721776665386756877, −2.00356841190108365448475978284, 0.67242270674577860728007482688, 1.83899597315427692437709961449, 2.69392964552538853368720980538, 3.61494945956080402827018731111, 5.59000565686395762544604389079, 6.49564191613541507662759976930, 7.43053261115471269276006664250, 8.070810914948245347271470629649, 8.494069972947517754401801463213, 9.498010999719590744897860661645

Graph of the ZZ-function along the critical line