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Properties

Label 2-980-28.27-c1-0-29
Degree 22
Conductor 980980
Sign 0.475+0.879i0.475 + 0.879i
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  + (−1.40 − 0.153i)2-s − 3.02·3-s + (1.95 + 0.432i)4-s − i·5-s + (4.25 + 0.465i)6-s + (−2.67 − 0.908i)8-s + 6.16·9-s + (−0.153 + 1.40i)10-s + 1.19i·11-s + (−5.91 − 1.30i)12-s − 4.83i·13-s + 3.02i·15-s + (3.62 + 1.68i)16-s + 2.54i·17-s + (−8.66 − 0.948i)18-s + 1.42·19-s + ⋯
L(s)  = 1  + (−0.994 − 0.108i)2-s − 1.74·3-s + (0.976 + 0.216i)4-s − 0.447i·5-s + (1.73 + 0.190i)6-s + (−0.946 − 0.321i)8-s + 2.05·9-s + (−0.0486 + 0.444i)10-s + 0.360i·11-s + (−1.70 − 0.378i)12-s − 1.34i·13-s + 0.781i·15-s + (0.906 + 0.422i)16-s + 0.617i·17-s + (−2.04 − 0.223i)18-s + 0.326·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.475+0.879i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.475+0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.475+0.879i0.475 + 0.879i
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), 0.475+0.879i)(2,\ 980,\ (\ :1/2),\ 0.475 + 0.879i)

Particular Values

L(1)L(1) ≈\approx 0.386082−0.230153i0.386082 - 0.230153i
L(12)L(\frac12) ≈\approx 0.386082−0.230153i0.386082 - 0.230153i
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+(1.40+0.153i)T 1 + (1.40 + 0.153i)T
5 1+iT 1 + iT
7 1 1
good3 1+3.02T+3T2 1 + 3.02T + 3T^{2}
11 1−1.19iT−11T2 1 - 1.19iT - 11T^{2}
13 1+4.83iT−13T2 1 + 4.83iT - 13T^{2}
17 1−2.54iT−17T2 1 - 2.54iT - 17T^{2}
19 1−1.42T+19T2 1 - 1.42T + 19T^{2}
23 1−5.80iT−23T2 1 - 5.80iT - 23T^{2}
29 1−0.774T+29T2 1 - 0.774T + 29T^{2}
31 1+6.63T+31T2 1 + 6.63T + 31T^{2}
37 1−5.10T+37T2 1 - 5.10T + 37T^{2}
41 1−7.46iT−41T2 1 - 7.46iT - 41T^{2}
43 1−1.38iT−43T2 1 - 1.38iT - 43T^{2}
47 1+1.07T+47T2 1 + 1.07T + 47T^{2}
53 1−3.36T+53T2 1 - 3.36T + 53T^{2}
59 1−9.88T+59T2 1 - 9.88T + 59T^{2}
61 1+9.59iT−61T2 1 + 9.59iT - 61T^{2}
67 1+10.5iT−67T2 1 + 10.5iT - 67T^{2}
71 1+16.3iT−71T2 1 + 16.3iT - 71T^{2}
73 1−0.107iT−73T2 1 - 0.107iT - 73T^{2}
79 1+10.7iT−79T2 1 + 10.7iT - 79T^{2}
83 1−15.8T+83T2 1 - 15.8T + 83T^{2}
89 1−3.94iT−89T2 1 - 3.94iT - 89T^{2}
97 1+8.71iT−97T2 1 + 8.71iT - 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

−9.979580087196258398308353148089, −9.314912054397237174605545344245, −8.017650538476340565726848960437, −7.43389568244093916382121492152, −6.36675297799163314957922500135, −5.70785623240316560587641706742, −4.91988416082240105923491440859, −3.48713887503132056797990642323, −1.66655205768209936100384408262, −0.52970971591702678161680242371, 0.853091938055869321148336069514, 2.30569928391142649297529787300, 4.04635796927074233221216909370, 5.27924874069704078107152294118, 6.03742206803704338360754631077, 6.88624741141006407409826161059, 7.21819002777721775289708802232, 8.592384535622678917319266606737, 9.499312936267187978274478790564, 10.28718837474594195019096775498

Graph of the ZZ-function along the critical line