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Properties

Label 2-990-33.32-c1-0-2
Degree 22
Conductor 990990
Sign 0.351−0.936i0.351 - 0.936i
Analytic cond. 7.905187.90518
Root an. cond. 2.811612.81161
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  − 2-s + 4-s − i·5-s − 2.65i·7-s − 8-s + i·10-s + (−2.74 + 1.86i)11-s + 7.17i·13-s + 2.65i·14-s + 16-s − 4.82·17-s + 0.755i·19-s − i·20-s + (2.74 − 1.86i)22-s + 7.83i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447i·5-s − 1.00i·7-s − 0.353·8-s + 0.316i·10-s + (−0.827 + 0.561i)11-s + 1.98i·13-s + 0.710i·14-s + 0.250·16-s − 1.17·17-s + 0.173i·19-s − 0.223i·20-s + (0.584 − 0.397i)22-s + 1.63i·23-s + ⋯

Functional equation

Λ(s)=(990s/2 ΓC(s) L(s)=((0.351−0.936i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(990s/2 ΓC(s+1/2) L(s)=((0.351−0.936i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 990990    =    2⋅32⋅5⋅112 \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.351−0.936i0.351 - 0.936i
Analytic conductor: 7.905187.90518
Root analytic conductor: 2.811612.81161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ990(791,⋅)\chi_{990} (791, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 990, ( :1/2), 0.351−0.936i)(2,\ 990,\ (\ :1/2),\ 0.351 - 0.936i)

Particular Values

L(1)L(1) ≈\approx 0.650638+0.450953i0.650638 + 0.450953i
L(12)L(\frac12) ≈\approx 0.650638+0.450953i0.650638 + 0.450953i
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+T 1 + T
3 1 1
5 1+iT 1 + iT
11 1+(2.74−1.86i)T 1 + (2.74 - 1.86i)T
good7 1+2.65iT−7T2 1 + 2.65iT - 7T^{2}
13 1−7.17iT−13T2 1 - 7.17iT - 13T^{2}
17 1+4.82T+17T2 1 + 4.82T + 17T^{2}
19 1−0.755iT−19T2 1 - 0.755iT - 19T^{2}
23 1−7.83iT−23T2 1 - 7.83iT - 23T^{2}
29 1−3.75T+29T2 1 - 3.75T + 29T^{2}
31 1−5.75T+31T2 1 - 5.75T + 31T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1−3.92T+41T2 1 - 3.92T + 41T^{2}
43 1+2.07iT−43T2 1 + 2.07iT - 43T^{2}
47 1−8.07iT−47T2 1 - 8.07iT - 47T^{2}
53 1+3.55iT−53T2 1 + 3.55iT - 53T^{2}
59 1+8.41iT−59T2 1 + 8.41iT - 59T^{2}
61 1−9.72iT−61T2 1 - 9.72iT - 61T^{2}
67 1+13.4T+67T2 1 + 13.4T + 67T^{2}
71 1+2.17iT−71T2 1 + 2.17iT - 71T^{2}
73 1−7.75iT−73T2 1 - 7.75iT - 73T^{2}
79 1−4.58iT−79T2 1 - 4.58iT - 79T^{2}
83 1+6.72T+83T2 1 + 6.72T + 83T^{2}
89 1−10.4iT−89T2 1 - 10.4iT - 89T^{2}
97 1+15.3T+97T2 1 + 15.3T + 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

−9.895691957725307157357648004570, −9.452841362094192413345825349154, −8.547190602698300232879208768057, −7.60485330753149111793101889216, −7.00688887495891682059687304375, −6.11064432371435588215087756508, −4.68564553100735863492217598611, −4.09433088161964445099026593853, −2.45356787564394560523801387518, −1.30139547143764523110528677050, 0.48276501800401486827803001585, 2.58654373685677541256496386481, 2.85275326564997821076252045390, 4.65268888974632945287253736229, 5.79056069276133141759178151024, 6.33085986921118478027619181227, 7.54573261305548565901528587860, 8.329882235321117927176568355317, 8.768186495116050188744284059696, 9.962692383734088161617816470989

Graph of the ZZ-function along the critical line