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Properties

Label 2-980-140.139-c1-0-99
Degree 22
Conductor 980980
Sign −0.422−0.906i-0.422 - 0.906i
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  + (−1.14 − 0.826i)2-s − 1.52i·3-s + (0.632 + 1.89i)4-s + (0.0967 − 2.23i)5-s + (−1.25 + 1.74i)6-s + (0.843 − 2.69i)8-s + 0.679·9-s + (−1.95 + 2.48i)10-s + 4.56i·11-s + (2.89 − 0.963i)12-s − 2.19·13-s + (−3.40 − 0.147i)15-s + (−3.19 + 2.40i)16-s − 6.22·17-s + (−0.779 − 0.561i)18-s − 3.83·19-s + ⋯
L(s)  = 1  + (−0.811 − 0.584i)2-s − 0.879i·3-s + (0.316 + 0.948i)4-s + (0.0432 − 0.999i)5-s + (−0.514 + 0.713i)6-s + (0.298 − 0.954i)8-s + 0.226·9-s + (−0.619 + 0.785i)10-s + 1.37i·11-s + (0.834 − 0.278i)12-s − 0.608·13-s + (−0.878 − 0.0380i)15-s + (−0.799 + 0.600i)16-s − 1.50·17-s + (−0.183 − 0.132i)18-s − 0.879·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.422−0.906i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.422−0.906i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.422−0.906i-0.422 - 0.906i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(979,⋅)\chi_{980} (979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.422−0.906i)(2,\ 980,\ (\ :1/2),\ -0.422 - 0.906i)

Particular Values

L(1)L(1) ≈\approx 0.125837+0.197398i0.125837 + 0.197398i
L(12)L(\frac12) ≈\approx 0.125837+0.197398i0.125837 + 0.197398i
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+(1.14+0.826i)T 1 + (1.14 + 0.826i)T
5 1+(−0.0967+2.23i)T 1 + (-0.0967 + 2.23i)T
7 1 1
good3 1+1.52iT−3T2 1 + 1.52iT - 3T^{2}
11 1−4.56iT−11T2 1 - 4.56iT - 11T^{2}
13 1+2.19T+13T2 1 + 2.19T + 13T^{2}
17 1+6.22T+17T2 1 + 6.22T + 17T^{2}
19 1+3.83T+19T2 1 + 3.83T + 19T^{2}
23 1+0.430T+23T2 1 + 0.430T + 23T^{2}
29 1+0.473T+29T2 1 + 0.473T + 29T^{2}
31 1+7.59T+31T2 1 + 7.59T + 31T^{2}
37 1+8.44iT−37T2 1 + 8.44iT - 37T^{2}
41 1+1.45iT−41T2 1 + 1.45iT - 41T^{2}
43 1+8.58T+43T2 1 + 8.58T + 43T^{2}
47 1−4.48iT−47T2 1 - 4.48iT - 47T^{2}
53 1−9.23iT−53T2 1 - 9.23iT - 53T^{2}
59 1−3.13T+59T2 1 - 3.13T + 59T^{2}
61 1+5.71iT−61T2 1 + 5.71iT - 61T^{2}
67 1+14.9T+67T2 1 + 14.9T + 67T^{2}
71 1+4.57iT−71T2 1 + 4.57iT - 71T^{2}
73 1−12.2T+73T2 1 - 12.2T + 73T^{2}
79 1+6.20iT−79T2 1 + 6.20iT - 79T^{2}
83 1+7.69iT−83T2 1 + 7.69iT - 83T^{2}
89 1−9.32iT−89T2 1 - 9.32iT - 89T^{2}
97 1+9.05T+97T2 1 + 9.05T + 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

−9.314974476410798558292259893383, −8.778265275489245731878957699367, −7.70874185771264028607160162006, −7.22984499001085430621457423945, −6.34309793249331680148470293743, −4.76968991588178773404521646430, −4.07824808300764562922634013702, −2.19624139128587604962509266734, −1.75053356044522300717425707235, −0.13068561236500856471505286186, 2.06034527519330873109769291952, 3.34137533968933603485512149029, 4.51449736796403513373208123139, 5.56284275249830187530358678225, 6.54255554473347296490868519582, 7.07242693405467271174657273243, 8.248640082670736317885814853158, 8.901059161446712411563714098422, 9.803565098047109107924395206591, 10.40256147771747924780445456387

Graph of the ZZ-function along the critical line