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Properties

Label 2-980-28.27-c1-0-57
Degree 22
Conductor 980980
Sign 0.381+0.924i0.381 + 0.924i
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  + (1.39 + 0.226i)2-s − 1.79·3-s + (1.89 + 0.632i)4-s − i·5-s + (−2.49 − 0.405i)6-s + (2.50 + 1.31i)8-s + 0.206·9-s + (0.226 − 1.39i)10-s − 4.23i·11-s + (−3.39 − 1.13i)12-s − 2.98i·13-s + 1.79i·15-s + (3.19 + 2.40i)16-s − 2.21i·17-s + (0.288 + 0.0468i)18-s − 4.56·19-s + ⋯
L(s)  = 1  + (0.987 + 0.160i)2-s − 1.03·3-s + (0.948 + 0.316i)4-s − 0.447i·5-s + (−1.02 − 0.165i)6-s + (0.885 + 0.464i)8-s + 0.0689·9-s + (0.0716 − 0.441i)10-s − 1.27i·11-s + (−0.980 − 0.327i)12-s − 0.827i·13-s + 0.462i·15-s + (0.799 + 0.600i)16-s − 0.537i·17-s + (0.0680 + 0.0110i)18-s − 1.04·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.381+0.924i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.381+0.924i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.381+0.924i0.381 + 0.924i
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), 0.381+0.924i)(2,\ 980,\ (\ :1/2),\ 0.381 + 0.924i)

Particular Values

L(1)L(1) ≈\approx 1.51216−1.01142i1.51216 - 1.01142i
L(12)L(\frac12) ≈\approx 1.51216−1.01142i1.51216 - 1.01142i
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+(−1.39−0.226i)T 1 + (-1.39 - 0.226i)T
5 1+iT 1 + iT
7 1 1
good3 1+1.79T+3T2 1 + 1.79T + 3T^{2}
11 1+4.23iT−11T2 1 + 4.23iT - 11T^{2}
13 1+2.98iT−13T2 1 + 2.98iT - 13T^{2}
17 1+2.21iT−17T2 1 + 2.21iT - 17T^{2}
19 1+4.56T+19T2 1 + 4.56T + 19T^{2}
23 1+2.05iT−23T2 1 + 2.05iT - 23T^{2}
29 1−6.42T+29T2 1 - 6.42T + 29T^{2}
31 1−2.40T+31T2 1 - 2.40T + 31T^{2}
37 1+4.32T+37T2 1 + 4.32T + 37T^{2}
41 1+4.88iT−41T2 1 + 4.88iT - 41T^{2}
43 1+12.3iT−43T2 1 + 12.3iT - 43T^{2}
47 1−6.76T+47T2 1 - 6.76T + 47T^{2}
53 1+12.8T+53T2 1 + 12.8T + 53T^{2}
59 1−13.9T+59T2 1 - 13.9T + 59T^{2}
61 1+0.0226iT−61T2 1 + 0.0226iT - 61T^{2}
67 1+5.06iT−67T2 1 + 5.06iT - 67T^{2}
71 1−4.07iT−71T2 1 - 4.07iT - 71T^{2}
73 1+3.33iT−73T2 1 + 3.33iT - 73T^{2}
79 1+3.62iT−79T2 1 + 3.62iT - 79T^{2}
83 1+11.7T+83T2 1 + 11.7T + 83T^{2}
89 1−16.6iT−89T2 1 - 16.6iT - 89T^{2}
97 1−12.0iT−97T2 1 - 12.0iT - 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

−10.39068338575000764683541923510, −8.738055548803464995958119190575, −8.170259809720899703315708461253, −6.91371453700344662864901152574, −6.15970526938683559674878080026, −5.48353481354280550800650916691, −4.83658603061008165604712878334, −3.68627202431535048534483963251, −2.56017685334787220028645031749, −0.68924937456661831979668879692, 1.65841459318702703338343320756, 2.83952580304490974326654684108, 4.26533076194721281399144350685, 4.76915837385436204244085073526, 5.90899891678267950857619664071, 6.53772971099338622716181743959, 7.14395508142578620562518687831, 8.347475558408541799108780353196, 9.775090377055800434000253901143, 10.37930904017656657574599863222

Graph of the ZZ-function along the critical line