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Properties

Label 2-799-799.798-c0-0-0
Degree 22
Conductor 799799
Sign −0.809−0.587i-0.809 - 0.587i
Analytic cond. 0.3987520.398752
Root an. cond. 0.6314680.631468
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s + 1.17i·3-s − 0.618·4-s − 0.726i·6-s + 1.90i·7-s + 8-s − 0.381·9-s − 0.726i·12-s − 1.17i·14-s + (−0.809 − 0.587i)17-s + 0.236·18-s − 2.23·21-s + 1.17i·24-s − 25-s + 0.726i·27-s − 1.17i·28-s + ⋯
L(s)  = 1  − 0.618·2-s + 1.17i·3-s − 0.618·4-s − 0.726i·6-s + 1.90i·7-s + 8-s − 0.381·9-s − 0.726i·12-s − 1.17i·14-s + (−0.809 − 0.587i)17-s + 0.236·18-s − 2.23·21-s + 1.17i·24-s − 25-s + 0.726i·27-s − 1.17i·28-s + ⋯

Functional equation

Λ(s)=(799s/2 ΓC(s) L(s)=((−0.809−0.587i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(799s/2 ΓC(s) L(s)=((−0.809−0.587i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 799799    =    17⋅4717 \cdot 47
Sign: −0.809−0.587i-0.809 - 0.587i
Analytic conductor: 0.3987520.398752
Root analytic conductor: 0.6314680.631468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ799(798,⋅)\chi_{799} (798, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 799, ( :0), −0.809−0.587i)(2,\ 799,\ (\ :0),\ -0.809 - 0.587i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.54373778240.5437377824
L(12)L(\frac12) ≈\approx 0.54373778240.5437377824
L(1)L(1) 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)
bad17 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
47 1−T 1 - T
good2 1+0.618T+T2 1 + 0.618T + T^{2}
3 1−1.17iT−T2 1 - 1.17iT - T^{2}
5 1+T2 1 + T^{2}
7 1−1.90iT−T2 1 - 1.90iT - T^{2}
11 1+T2 1 + T^{2}
13 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1−1.17iT−T2 1 - 1.17iT - T^{2}
41 1+T2 1 + T^{2}
43 1−T2 1 - T^{2}
53 1−0.618T+T2 1 - 0.618T + T^{2}
59 1+0.618T+T2 1 + 0.618T + T^{2}
61 1+1.90iT−T2 1 + 1.90iT - T^{2}
67 1−T2 1 - T^{2}
71 1+1.17iT−T2 1 + 1.17iT - T^{2}
73 1+T2 1 + T^{2}
79 1−1.17iT−T2 1 - 1.17iT - T^{2}
83 1−2T+T2 1 - 2T + T^{2}
89 1−0.618T+T2 1 - 0.618T + T^{2}
97 1−1.90iT−T2 1 - 1.90iT - T^{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

−10.55659384046715586786129192363, −9.658503490873419471616398030091, −9.230608660697126333432813215695, −8.668012208593368509539302893483, −7.77272380949124463408614760904, −6.30859000117540929748295568381, −5.22943843399799874715578687605, −4.70331704210385831845864675399, −3.51475578958502495483302169726, −2.15833606164075396586359970695, 0.73193377152842458121715506240, 1.85968381314560660724426055721, 3.84099739825948163578429689560, 4.48453321000380256502910651092, 6.01147990756952397556880159451, 7.12765120247226943879114138473, 7.47261228052334532118473597486, 8.249419379799970576825237215070, 9.224021066914751676971607364408, 10.26015650052542518179465895946

Graph of the ZZ-function along the critical line