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Properties

Label 2-980-28.27-c1-0-50
Degree 22
Conductor 980980
Sign 0.860−0.509i0.860 - 0.509i
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)=((0.860−0.509i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.860−0.509i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.860−0.509i0.860 - 0.509i
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), 0.860−0.509i)(2,\ 980,\ (\ :1/2),\ 0.860 - 0.509i)

Particular Values

L(1)L(1) ≈\approx 3.96824+1.08780i3.96824 + 1.08780i
L(12)L(\frac12) ≈\approx 3.96824+1.08780i3.96824 + 1.08780i
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

−10.16764493346148005069741531365, −9.017365630627522458215342514510, −8.314549720793111438702563586632, −7.54232001329272183397345480185, −6.55687832398514942469006964213, −5.83457806615879956443027780755, −4.68448410517532051367686299802, −3.46145819099473033641962933345, −3.09038327291096125125521203664, −1.85830878008574147021592834847, 1.54981500922309939627542348029, 2.73764181455946898719105299961, 3.44940502504428906989669175258, 4.65862194989030031197992411903, 5.29018292342399297706462270067, 6.46936660953977215787787552530, 7.56775209278982203751034904757, 7.978878805386600797134541033374, 9.299626347242446577075661180804, 9.790949367706515258173616923898

Graph of the ZZ-function along the critical line